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A unified description of classical and quantal scattering Turner, Ralph Eric 1978

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A UNIFIED DESCRIPTION OF CLASSICAL AND QUANTAL SCATTERING by RALPH ERIC TURNER B. Sc. (Hons.) University of Waterloo, 1 9 7 2 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (CHEMISTRY) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1 9 7 8 © R a l p h E r i c Turner, 1 9 7 8 In presenting t h i s thesis i n p a r t i a l f u l f i l l m e n t of the requirements f o r an advanced degree at the University of B r i t i s h Columbia,I agree that the Library s h a l l make i t f r e e l y available f o r reference and study. I further agree that permission f o r extensive copying of t h i s thesis f o r scholarly purposes may be granted by the Head of my Department or by his representatives. I t i s understood that copying or publication of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Chemistry The University of B r i t i s h Columbia Vancouver,British Columbia,Canada V6T 1W5 ABSTAGT A unified description of classical and quantal scattering i s presented. This description i s unified both in the mathematical language and in the conceptual pictures that are used. S t a t i s t i c a l states and observables,repre-sented i n phase space,are emphasized while both the time dependent trajectory and stationary state beam pictures are used. Semiclassical-type approximation schemes to the generalized differential cross section are presented for the single and double potential cases. Connection i s made with the Born and distorted wave Born approximations,for both quantum and classical mechanics. A similar semiclass-ical-type approximation scheme for the double potential internal state kinetic cross section i s presented. Connection i s also made with the 'constant acceleration approximation* of Oppenheim and Bloom. Zwanzig's projection operator method i s used to express the kinetic cross sections i n terms of memory effects for the interacting observables. i i i TABLE OF CONTENTS Abstract i i Table of Contents i i i L i s t of Tables v i i L i s t of Figures v i i i Acknowledgements i x CHAPTER 1 Introduction 1 CHAPTER 2 Phase Space Representation f o r C l a s s i c a l and Quantum Scattering 7 2.1 Introduction 8 2.2 Phase Space Representation 11 2.2.a Formal Structure 11 2.2.b C l a s s i c a l Mechanics 16 2.2.c Quantum Mechanics 18 2.2.d The Flux Observable 23 2.2. e Summary 25 2.3 Superoperators as Phase Space Transformations ..26 2.3. a General Structure 26 2.3.b C l a s s i c a l Mechanics 30 2.3.C Quantum Mechanics 33 2,k Aspects of Free Motion 38 2.5 Interacting Motion 50 2.6 The M i l l e r Superoperator ..60 CHAPTER 3 Scattering as a Time Dependent Process ... 75 3.1 Introduction 76 iv. 3.2 The Collision Process 81 3.3 Cross Sections 88 3.4 Superoperator Formulas for the Cross Section ...92 3.5 Discussion 99 CHAPTER 4 Classical Trajectory Approximation 102 4.1 Introduction 103 4.2 Correlation Function Formulation 109 4.3 Exact Classical Cross Section 114 4.4 Ex p l i c i t CTA Cross Section 118 4.5 Straight Line Trajectory Approximation and the Born Cross Section 127 4.5.a The SLCTA 128 4.5«"b Classical Born Approximation 132 4.5.c Semiclassical Mott Formula 139 4.6 Discussion 141 CHAPTER 5 Double Potential Scattering:Distorted Wave Born and Semiclassical-type Approximations 145 5.1 Introduction 146 5.2 Miller and Transition Superoperators for Double Potentials 150 5.2.a Abstract and Parameterized Superoperators 150 5.2.b Miller Superoperator 152 5.2.c Transition Superoperator 156 5.3 Cross Section 160 5.4 Quantal DWBA 164 V. 5.5 Classical Cross Section and Classical DWBA 170 5»5»a Exact Classical Cross Section 170 5.5«D Reference Trajectory Approximation:The Classical DWBA 177 5.6 Static and Dynamic Interference Approximations .180 5.7 Classical Trajectory Approximation ....183 5.8 Discussion 187 CHAPTER 6 Kinetic Cross SectionssTrajectory Approximations 189 6.1 Introduction 190 6.2 Kinetic Cross Sections 193 6.3 Double Potential Formalism 19? 6.4 Phase Space Representation of the B.. Operators 203 6.5 Semiclassical-type Approximations and the Distorted Wave Born Approximation 209 6.5»a Distorted Wave Born Approximation 209 6.5»D Semiclassical-type Approximations 213 6.6 KDWFCAfKDWCTA and Straight Line Approximations .217 6.6.a KDWFCA and KDWCTA 217 6.6.b Straight Line Approximations 219 6.7 Discussion .....221 CHAPTER 7 Kinetic Cross Sections Projection Techniques 223 7.1 Introduction 22k 7.2 General Projection Formulae 226 7.2.a The One Dimensional Case .229 7.2.b The Two Dimensional Case 230 v i . 7.2.c Equivalence of the Memory Expression to the Original Projected Resolvent 232 7.3 Relation to the Semiclassical-type Approximations 235 7.4 Resolvent Approximations 240 7.4.a Importance of the (1 - 6 M Space Motion ....240 7.4. b Resolvent Approximations 242 7.5 Comments on the Eigenfunctions of (1-CP ) ^ t r ( l - C P ) 244 7.5. a M/ller Superoperator 244 7.5.D Separable Nature of V^> 248 7.5.C Eigenfunctions 250 7.6 Discussion 253 CHAPTER 8 Conclusion and Directions for Future Work 256 Bibliography 262 Appendix A 272 v i i . LIST OF TABLES Table 4.1 142 Table 6.1 222 Table 7.1 25^ v i i i . LIST OF FIGURES Fig. 2.1 Geometry of a Hard Sphere Collision 67 Fig. 2.2 Individual Classical Trajectories in Hard Sphere Scattering 70 Fig. 2.3 Classical Streamlines for Hard Sphere Scattering 72 Fig. 3.1 Trajectory for the Center of the Phase Space Packet 84 Fig. 4.1 CTA and SLCTA Trajectories 129 Fig. 4.2 Exact Classical and Straight Line Trajectories 133 Fig. 5.1 The Trajectory R*(t|r,pR) 173 ix. ACKNOWLEDGEMENTS I would like to thank Dr. Robert F. Snider for his supervision of the years of work that culminated in the writing of this thesis. Without his support and openness this work would never have been completed. I would also like to thank a l l the graduate students and post doctoral fellows of room 126Y for the many f r u i t f u l discussions of the past few years. As well,I would also like to thank the University for i t s financial support. C H A P T E R 1 I N T R O D U C T I O N 2 . 1.INTRODUCTION Of primary importance in physical theory i s the calculation of expectation values of physical observables. A particular example i s the particle flux (number of particles per unit area per unit time) undergoing binary collisions. Rather than emphasizing the particle flux directly,the ratio of outgoing to incoming particle fluxes defines the differential cross section. There are two conceptual pictures of the co l l i s i o n event{namely, the time dependent trajectory and stationary state beam pictures. As well,the underlying dynamics i s either classical or quantal. i Por classical mechanics ,a s t a t i s t i c a l state i s represented by a distribution function i n phase space while a physical observable i s a phase space function. In the description of the scattering phenomena,the incoming classical s t a t i s t i c a l state i s standardly 1 taken to be pure{that i s , a product of a Dirae delta function i n position and a Dirac delta function i n momentum. This pure classical state represents exact knowledge of the relative position and momentum of the colliding particles?that is,the particles are localized at phase points. In this s t a t i s t i c a l state, the expectation value of a physical observable i s equal to the phase space function representing that physical observable. The dynamical time dependence of the expectation value i s then the same as the time dependence of a physical observable;this i s the Heisenberg picture. In terms of the 3. relative coordinate phase point to phase point trajectories of the position and momentum observables,the classical differential cross section i s the ratio of the number of trajectories per unit steradian exiting i n a particular direction at a position outside the range of the potential to the number of entering trajectories per unit area passing through a plane perpendicular to the incoming momentum. The standard classical scattering theory* thus deals with physical observables as phase space functions whose co l l i s i o n a l changes are obtained by following the time dependence of classical trajectories. 2 In the quantal case ,the physical observables are standardly represented by operators on a Hilbert space while the set of possible s t a t i s t i c a l states are those operators whose dual i s the space of observable,operators. If the observable operators are bounded,then the s t a t i s t i c a l states are trace class operators (density operators). For scattering theory,this s t a t i s t i c a l state i s standardly^' * J* 6 7 taken to be pure. Since a pure state i s equivalent to a Hilbert space element,then quantal c o l l i s i o n theory can be developed entirely i n terms of Hilbert space elements. o In contrast,the work of Snider develops the quantal differential cross section from the density operator view-point. In the usual Hilbert space approach,the dynamical time dependence i s governed by the time dependence of the wavefunctionsjthis i s the Schrodinger picture. Rather than dealing with the time dependence directly,stationary state 4. 3 4 6 7 wave functions are usually^' ' *' considered,contrast Taylor's-' time dependent wave function approach. The quantal differential cross section i s defined by the ratio of the outgoing spherical stationary state flux to the incoming planar stationary state flux. In this way, quantal scattering theory deals with stationary state wave functions having the conceptual interpretation of being equivalent to the picture of stationary state beam scattering. Not only i s the mathematical description of classical and quantal collisions different;the former emphasizing phase space observables while the latter emphasizes station-ary state wave functions,but the conceptual pictures also differ;time dependent partiele trajectories versus a stationary state beam (wave motion). It i s the purpose of this thesis to describe classical and quantal scattering events i n the same mathematical language and with the same conceptual pictures. To achieve the former,both classical and quantum mechanics are written in a form involving s t a t i s t i c a l states and observables. This i s reviewed in chapter 2,using phase space as a common language of represen-tation. This i s the normal classical representation while o quantally,the Weyl correspondence^ i s used to make a connection with the usual representation of quantal obser-vables and density operators. Of the many equivalent phase space representations 1 0,the Weyl correspondence i s chosen because i n this representation many quantal observables are equal to their classical counterparts and because the 5. i i Wigner function , the Weyl correspondence of the density operator,has the correct marginal distribution functions. A moment method for the time dependence of the average position and momentum,valid both classically and quantally, i s presented and compared to the time dependent Gaussian 1 2 wave packet approximation of Heller . In chapter 3»an expression for the differential cross section,Eq. (3«4.15)» i s developed which i s applicable for both classical and quantum mechanics. It i s derived by means of both conceptual pictures;that is,the time dependent trajectory and stationary state beam pictures. Having an expression valid for both mechanics,the same types of approximations can be applied for both mechanics. For example,the Born and distorted wave Born approximations to the quantal and classical situations are compared. As well,semiclassical-type approximation schemes are presented. The relation of these semiclassical-type approximations to the standard semiclassical approx-13 14 imations of,say,Marcus J and Miller i s yet to be eluci-dated. It i s hoped that the present formalism w i l l lead to a better understanding of semiclassical approximations. Kinetic cross sections (that is,cross sections averaged over the Maxwellian velocity distribution) can also be defined i n a language which i s valid for both mechanics. In chapter 6,a semiclassical-type trajectory approximation scheme to the internal state kinetic cross sections i s presented and related to the 'constant acceleration approximation* of Oppenheim and Bloom - 0. 16 Zwanzig's projection operator method i s applied to the k i n e t i c cross sections i n chapter ?. F i n a l l y , t h e semi-c l a s s i c a l - t y p e approximation scheme of chapter 6 i s related to the projection operator method. CHAPTER 2 PHASE SPACE REPRESENTATION FOR CLASSICAL AND QUANTUM SCATTERING 8. 2.1.INTRODUCTION C l a s s i c a l and quantum mechanics are usually formulated using very d i f f e r e n t mathematics, the f i r s t s t r e s s i n g phase space functions while the l a t t e r involves wavefunctions i n a H i l b e r t space. Such d i f f e r e n t formulations emphasize the differences between c l a s s i c a l and quantum mechanics, so much so that t h e i r s i m i l a r i t i e s are often l o s t . I t i s the object of the f i r s t two chapters of t h i s thesis to emphasize the s i m i l a r i t i e s between c l a s s i c a l and quantum mechanics, with the s p e c i f i c goal of describing c o l l i s i o n phenomena -i n a manner which has a meaning i n both mechanics. By t h i s means, i t i s possible to immediately see what i s the c l a s s i c a l l i m i t (provided i t exists) of a quantum cross section. Hopefully, t h i s w i l l lead to a better understanding of semiclassical approximations. To emphasize the s i m i l a r i t i e s between the two mechanics, a u n i f i e d mathematical formulation i s required. This u n i f i c a t i o n may be done i n two ways; either formulate c l a s s i c a l mechanics i n terms of p r o b a b i l i t y amplitudes which may be compared with the usual quantum wavefunctions, or formulate quantum mechanics i n phase space, which may then be compared with the usual description of c l a s s i c a l mechanics. v 1 The f i r s t method has recently been proposed by Prugovecki , while i t i s the l a t t e r method that i s described here. The l a t t e r method i s preferred here because i t nat u r a l l y stresses physical observables and s t a t i s t i c a l states, while eliminating any dependence on unobservable quantities ( i . e . wavefunctions). 2 Since the quantum Wigner function and the coherent state representation of the density operator (used by Prugovecki ) are equivalent, then any expectation value may be calculated using either representation to give the same r e s u l t , provided that the associated observable i s also transcribed into the appropriate representation. The representation involving the Wigner function i s preferred here because i t gives the correct marginal d i s t r i b u t i o n functions (of position density and momentum probability) whereas the coherent state representation does not^. As well, the dynamic f o r the Wigner function i s simpler, e s p e c i a l l y f o r free motion 1. The description of c l a s s i c a l and quantum observables i n a phase space representation i s the object of Sec .2V'2. D i s t i n c t i o n between observables and s t a t i s t i c a l states i s made throughout. This i s more necessary i n c l a s s i c a l mechanics than i n quantum mechanics. The connection between quantum operators and phase space functions i s made v i a the Weyl correspondence . Phase space transformations describe l i n e a r transformations of phase space functions. Sec.2,3 discusses the time evolution of states and observables as phase space transformations and f o r quantum systems, makes connections between these transformations and super-operators. Free motion (no poten t i a l acting on the p a r t i c l e ) and i n t e r a c t i n g motion are discussed i n Sees.2.4 and 2 . 5 . I t i s emphasized that there i s no difference between the free p a r t i c l e dynamics of a c l a s s i c a l and a quantal system, provided the c l a s s i c a l system i s s u f f i c i e n t l y "spread out". 10. A moment method for approximating the quantum dynamics of an interacting system i s discussed and compared with Heller's^ Gaussian wavepacket method. Finally, the Miller superoperator of c o l l i s i o n theory is expressed as a phase space transfor-mation,Sec.2.6. The formulation is exemplified by an explicit calculation for the classical scattering from a r i g i d sphere. In chapter 3 c o l l i s i o n theory is formulated as a time dependent phenomena, using a phase space representation so that i t i s valid for both classical and quantum mechanics. 2.2.PHASE SPACE REPRESENTATION  2.2.a.FORMAL STRUCTURE The emphasis i n this thesis i s on observables and s t a t i s t i c a l states. It seems appropriate then,to comment on the formal structure of physical theory,in order to at least explain the notation used i n the following. The discussion in chapters 2 and 3 i s restricted to one structureless particle moving i n 3-dimensional space,since this i s crucial to the description of the col l i s i o n processes. Internal states can, and will,be added later. The set of physical observables (either classically or quantally) i s usually assumed to have a vector space structure. In this thesis,the mathematical structure of the set of obser-vables i s not specified. Only the fact that the observables are elements of a vector space used. The set of observables allowed by classical mechanics and the set of observables allowed i n quantum mechanics are in general different,while many of the observables of physical interest are the same. By considering a l l possible observables,whether classical or quantal,to be elements of ^ ,both mechanics are included i n the same formalism. Notationally an observable A w i l l be formally ,the double ket notation following Baranger^ i n his description of spectroscopic phenomena i n "line space" or "Liouville space". Again this contrasts an element of & with a Dirac ket I ^ . Associated with the subset ^" c^ of & 0 f classical observables,is the set of classical states -C^^, which are the ( positive and normalized) linear functionals on 12. Analogously,the set of quantum states are the (positive, normalized) linear functionals on the set of quantum obser-vables A l l states can be incorporated into a vector space j*/* constructed as the vector space generated from the union of a n d d?Q* e xP e c^ a"ti° n value of an obser-vable A in state S i s then denoted by <A> S =X S ' A ^ = jC A | S > ¥' ( 2 - 2 > 1 ) which emphasizes that S i s an element ofj^, A an element of and that these spaces are i n general different, see in particular the classical case. A complex^conjugate has been introduced for convenience, that i s , observables that are measured are real, yet i t i s often useful to take complex combinations of these real observables (for example the raising and lowering operators in angular momentum theory) and use them as a basis for calculations. Equation (2.2.1) demonstrates the duality between whether the states are linear functionals on the set of observables, or vice versa. No further discussion of this point is given in- this chapter, but i t i s used in chapter 3. There are certain ideal elements in the spaces and .-• These are the phase points j r . p ^ a n d l^'P^j^* Formally, the observable | r,p^^asks whether the system is at the phase point r,p , while the state \r,p>S says that the system is at the phase point r,p. It i s not to be construed that these correspond to a physical observable and state, indeed they cannot i n quantum mechanics, yet their use as bases for O'a.nAj^is what is to be stressed. The 1 3 . normalization chosen for these two ideal elements i s //r,p I r\p^> = j k r - r ' j ^ p - p ' ) , ( 2 . 2 1 2 ) while completeness of the bases is expressed as and 1* = With the use of these ideal elements, the observable A i s represented by the phase space function a(r,p) =X 5 > E <2,2'5) while the state S i s represented by f(r.p) = \(r,P S > . ( 2 . 2 . 6 ) Completeness of the bases, Eqs. ( 2 . 2 . 3 - 4 ) , imply that the expectation value of Eq. ( 2 . 2 . 1 ) can be calculated as a phase space integral, namely = J^j"f*(r,p)a(r,p)drdp. ( 2 . 2 . 7 ) This has used the identity in $\ Eq. ( 2 . 2 . 4 ) , but a similar 14. expression, based on \ ', uses Eq. ( 2 . 2 . 3 ) to obtain the same result. Again f(r,p) i s correctly real, as i s a physical observable a(r,p), but the complex conjugate is carried along for completeness. It i s to be stressed that a(r,p) and f(r,p) represent, respectively, the observable A and the state S, while they in themselves may have no physical meaning; that i s , f(r,p) 6 can in general not be taken as the probability that the system in state S i s at point r,p. This i s because l r , p \ is not a physical observable (in quantum mechanics), yet i s s t i l l a valid basis for . That these are 1-1 represen-tations follows from the completeness arguments, which imply and fj ~Jjr fY/j S ??. = | | r,p>>, f(r,p)drdp. ( 2 . 2 . 9 ) The phase space function representing an observable has the dimensions of the observable. Thus the kinetic energy K(r,p) =^<^r,p|K^ = p2/2/<-, ( 2 . 2.10) and the potential energy V(r,p) = ( f ' r . p l v ^ = V(r) ( 2 . 2 . 1 1 ) observables have units of energy while the identity 15. ( 2 . 2 . 1 2 ) i s dimensionless. As an application of the latter, f(r,p) i s normalized 1 • • f(r,p)drdp. ( 2 . 2 . 1 3 ) Note that f(r,p) has the units of (action)"* 3, i.e., T3M"*3L~6, The specific properties of classical and quantum systems are described in turn. 1 6 . 2.2.b.CLASSICAL MECHANICS Classical quantities are standardly defined as phase space functions, so Eq. ( 2 . 2 . 7 ) i s more natural than i t s abstraction, Eq. ( 2 . 2 . 1 ) . The observation of the system at phase point r,p i s physically possible in classical mechanics. Thus the ideal element I r,p)& i s a valid observable, with the consequence that f(r,p) is the probability of observing the system at r,p. For this reason f(r,p) i s real and positive, a distribution in classical mechanics. It i s also allowed, that the system be at one phase point. Thus the ideal element /r*,p , ^ ? i s a physical state, whose phase space representative is f(r,p) = (fr,p | r \ p ^ = S(r-r') X( p-p«). (2.2.14) Such states are dispersion free and thus constitute the set of classically pure states. That i s , since the algebra of classical observables is commutative, and hence any observable is a phase space function, the dispersion of any observable A in the state Eq? ( 2 . 2 . 1 4 ) is zero, <AV = f(r,p)*a(r,p)drdp = a(r\p') ( 2 . 2 . 1 5 ) <^ (A - <A> S)/ > S =j Jf(r,p)*ra(r,p)-a(r',p ' )j 2drdp = 0. JJ ( 2 # 2 . 1 6 ) s well, any other real and positive function f(r,p) is a convex combination of such pure state functions, which means that the set of states / r ' » P ^ form the extreme set of the convex set of classical states. In classical mechanics there appears to be no natural space representations. A t r i v i a l difference is the differ-ence i n dimensionality, f(r,p) has something to do with probability, but to represent an observable probability (dimensionless), i t must be multiplied by the units of (action) 3. Such a quantity i s h 3 (Planck's constant), but this is non-classical, compare the quantum case. association between the spaces Of and 18. 2 ; 2.!c. QUANTUM .MECHANICS An observable A i n quantum mechanics i s standardly associated with an operator A on h i l b e r t space f^. An often used device, i s to consider only bounded operators, i n p a r t i c u l a r the projection operators, from which other quantities can be obtained. But with such a r e s t r i c t e d set of observables, the convex set of states are those positive (hermitian) operators of unit trace, the density operators J) *' The extreme points of t h i s set are the pure states, the projection operators J>f~ /¥><?/, (2.2.1?) while the expectation value of A i n state S^ i s given by the trace < A > s f - /s»lA^  • T =&kU*h ' [ T r A o P A J * - ^ j f A 0 p = <^ I Aop 1 ^ > • ( 2 - 2 - 1 8 ) written i n a v a r i e t y of forms. An a r b i t r a r y , state i s represented as a density operatorJ) which i s a l i n e a r combination of such pure states. One way of associating a phase space function with a 3 -quantum operator i s via the Weyl correspondence . This gives 1 9 . a 1-1 mapping "between hermitian operators and r e a l phase space functions. A useful quantity f o r making t h i s corres-pondence i s the (hermitian) operator A(r,p) = dq exp(-ip«q/n) 1 r - l q\/r+ _ r q / = j d P exp(ir.P/ f i ) | P - £ P ^ P + £ P | (2.2 .19) defined "by Leaf . The above d e f i n i t i o n involves Dirac ket-bra combinations i n position and momentum representations. Alternately matrix elements ofA(r,p) are / r ' ) A(r,p)|r") = e x p ( - i p - ( r " - r 1 )/fi) Sfr-ltir'+r" )J S < v 1 " ~ ' ~ / ^ ^ " L~ (2.2.20) ^P*/ A (r,p)| p"\ = expiir^p-'-p^ / f D a/p-iCp'+P")]. N - *• ~ ' ~ ' ~ ^ A, ^ "(2.2.21) These operators are orthogonal, and complete i n both phase space and operator space according to T r / U r , p ) A ( r \ p ' ) = h 3 $ ( r - r ' ) £(p-p') (2.2.22) and JJdrdp A (p.p) Tr[A ( r . p ) A o p ] . h \ p . (2.2.23) With the aid of ^ \ ( r , p ) , any observable A can be represented as a phase space function a(r.p) = <fr.p|A^= Tr A(j>p)A , (2.2."24) 2 0 . which can also be transcribed into an integral over matrix elements of A with the help of Eqs. (2.2.18) or ( 2 . 2 . 1 9 - 2 0 op - 2 1 ) . The inverse transformation follows from Eq. ( 2 , 2 . 2 3 ) as Ann = h" 3 / fdrdp A(r,p)a(r,p). ( 2 . 2 . 2 5 ) O p J J ~ «s ~ ~ ~ ^  In an analogous manner, the Wigner function f(r,p) which represents the state S (density o p e r a t o r ^ ) i s f(r,p) = ^ >p/S^, = h" 3Tr/\(r,p) J>% (2y2.26) with the inverse transformation P s = / fdrdp /6(r,p)f(r,p). ( 2 . 2 . 2 7 ) J J J ~ ~ ~ ~ On comparing the abstract formalism, Eqs. ( 2 . 2 . 2 ) to ( 2 . 2 . 9 ) , with their operator counterparts, Eqs. ( 2 . 2 . 2 2 ) to ( 2 . 2 . 2 7 ) . i t is seen that the associations | r , p ^ , * A ( r . p ) ( 2 . 2 . 2 8 ) and I r . p ^ ,p^> * h " 3 A(r,p) ( 2 . 2 . 2 9 ) determine the ideal elements for quantum mechanics. Since i t is the same operator ^  (r,p) that appears in both state and observable equivalences, there i s a natural association between state and observable spaces. This i s further 21. a c c o m p l i s h e d o n l y by the presence of a fun d a m e n t a l u n i t of a c t i o n , h. A p a r t i c u l a r example of the a s s o c i a t i o n between s t a t e s and o b s e r v a b l e s i s t o note t h a t h 3 f i s d i m e n s i o n l e s s and r e p r e s e n t s the q u e s t i o n o f whether the system i s i n s t a t e f . The answer i s , ( 2 . 2 . 3 0 ) which measures the p u r i t y o f the quantum s t a t e d 6 8 I t i s f a i r l y common '. -to s t r e s s the " s h o r t c o m i n g s " of the Wigner f u n c t i o n . That i s , the l i t e r a l i n t e r p r e t a t i o n of Eq. ( 2 . 2 . 2 6 ) i n l i g h t o f what an e x p e c t a t i o n v a l u e means, Eq. ( 2 . 2 . 1 ) , i m p l i e s t h a t f ( r , p ) i s t h e p r o b a b i l i t y t h a t t h e system i s a t p o s i t i o n r and momentum p. Wh i l e t h i s i s a v a l i d i n t e r p r e t a t i o n f o r c l a s s i c a l mechanics, i t i s not t r u e f o r quantum systems. I n p a r t i c u l a r , the quantum Wigner f u n c t i o n f ( r , p ) can be n e g a t i v e . F o r a pure s t a t e , L e a f ^ has shown t h a t f ( r , p ) i s the d i f f e r e n c e of p o s i t i v e f u n c t i o n s ff(r,p) = h"3jdq expt-ip.q/fij^ iql^ r-iq) - h"3 Jdq {f<r |=osf#q.(Pop-p)Allf>|2 -j<r|sin[|q.(pop-p)A]^)|2] , (2.2.3 1) which d i s a l l o w s i n t e r p r e t i n g f ( r , p ) as a d i s t r i b u t i o n 22. function. Yet a l l allowed expectation values can be calculated from f ( r , p ) . I t i s for, t h i s purpose that i t i s stressed that f ( r , p ) i s a representation of the quantum s t a t i s t i c a l state S (density o p e r a t o r ^ ) , just as the wavef unction ^ ( r ) = S^lp represents the abstract Dirac state . I t i s not the physics i n f ( r , p ) that i s wrong, but ~ A ^ rather i t i s the attempt to interpret the quantum f ( r , p ) as a d i s t r i b u t i o n function that i s incorrect. In the present formalism, t h i s i s traced back to the f a c t that the respective i d e a l elements |r, p 1 ^ and |r,p"^ are respec-t i v e l y , not allowed quantum observables or quantum states. 2 3 . 2.2.d.THE PARTICLE FLUX OBSERVABLE -H i s t o r i c a l l y , t he o p e r a t o r s i n quantum mechanics were o b t a i n e d from t h e i r c l a s s i c a l c o u n t e r p a r t s by a c o r r e s -pondence p r i n c i p l e ^ , f o r m a l i z e d by Eq. ( 2 . 2 . 2 5 ) . S i n c e t he s e t o f o b s e r v a b l e s i s u s u a l l y c o n s i d e r e d t o be an a l g e b r a , t h e r e a r e two laws of c o m p o s i t i o n . The f i r s t of thes e i s a d d i t i v i t y and t h i s i s o b v i o u s l y p r e s e r v e d by the c o r r e s -pondence. However, the m u l t i p l i c a t i o n laws a r e v e r y d i f f e r e n t . I n c l a s s i c a l mechanics, the phase space f u n c t i o n s r e p r e s e n t i n g the o b s e r v a b l e s a r e m u l t i p l i e d , whereas i n quantum mechanics, o p e r a t o r ( m a t r i x ) m u l t i p l i c a t i o n i s i n v o l v e d . When r e p r e s e n t e d i n phase space, t h i s o p e r a t o r m u l t i p l i c a t i o n i s e x c e e d i n g l y complex^. Simple o b s e r v a b l e s a r e the k i n e t i c and p o t e n t i a l e n e r g i e s and the i d e n t i t y , whose phase space r e p r e s e n t a t i o n s a r e g i v e n by Eqs. ( 2 . 2 . 1 0 - 1 1 - 1 2 ) . A n o t h e r s i m p l e o b s e r v a b l e i s the p a r t i c l e d e n s i t y »<S> -4l<S>>3 -f«5.E ) d£ f<RJf[*>)in quantum J mechanics ( 2 . 2 . 3 2 ) which i s r e p r e s e n t e d by the o p e r a t o r A ( r -R) i n the v /v op ~ u s u a l f o r m u l a t i o n of quantum mechanics, and by S(r-R) as A / i n the phase space r e p r e s e n t a t i o n of bo t h c l a s s i c a l and quantum mechanics. A more complex o b s e r v a b l e i s the p a r t i c l e f l u x . Here the phase space r e p r e s e n t a t i o n i s easy, namely J ( r , p | R ) = ( p > ) S(r-R), A / A / As A * A / J AJ A / ( 2 . 2 . 3 3 ) 24. with the expectation value This i s v a l i d f o r both c l a s s i c a l and quantum mechanics. In fa c t , the operator corresonding to J(R) i s obtained by Eq. ( 2 . 2 . 2 5 ) to be k [Bo//1** S^oV-Vl + > ( 2 . 2 . 3 5 ) J .(R) = -s ys/ Op A/ " which can also be obtained by other methods. 25. 2.2.e.SUMMARY I t has been s t r e s s e d t h a t b o t h c l a s s i c a l and quantum mechanics can be f o r m u l a t e d i n phase space w i t h many o b s e r v a b l e s ( l i k e l y a l l the ones of p h y s i c a l i n t e r e s t ) r e p r e s e n t e d by the same ( c l a s s i c a l ) f u n c t i o n . But the c l a s s o f a l l o w e d s t a t e s i s v e r y d i f f e r e n t . I n each c a s e , the s e t of s t a t e s i s convex, but t h e i n t e r s e c t i o n o f t h e two s e t s appears t o be v e r y s m a l l . That i s , the Wigner f u n c t i o n must be s p r e a d i n o r d e r t o s a t i s f y t he u n c e r t a i n t y p r i n c i p l e , and i s g e n e r a l l y n e g a t i v e , i n some p a r t of phase space, whereas c l a s s i c a l d i s t r i b u t i o n s must be p o s i t i v e , but n e g l i g i b l e s p r e a d s a r e a l l o w e d . F i n a l l y , i t i s remarked t h a t the c o n d i t i o n of b e i n g t r a c e c l a s s i n quantum mechanics, o r more g e n e r a l l y normed a c c o r d i n g t o Eq. (2.2.13) f o r both c l a s s i c a l and quantum mechanics, i s r e q u i r e d and s u f f i c i e n t i f a l l o b s e r v a b l e s a r e bounded. On the o t h e r hand, most o b s e r v a b l e s o f i n t e r e s t , i n p a r t i c u l a r the momentum and energy, a r e unbounded ( b o t h c l a s s i c a l l y and quantum m e c h a n i c a l l y ) . T h i s means t h a t t h e c l a s s o f a l l o w e d s t a t e s ( i n b o t h mechanics) must be f u r t h e r r e s t r i c t e d , u s u a l l y by some growth p r o p e r t i e s f o r l a r g e momenta i n the phase space r e p r e s e n t a t i o n . 26. 2.3.SUPEROPERATORS AS PHASE SPACE TRANSFORMATIONS  2.3.a.GENERAL STRUCTURE Linear transformations of states and observables are often required for the description of physical phenom-ena. Dynamical time evolution i s one example of such a transformation. The word superoperator was coined by 10 Crawford to distinquish an "operator on operators" from an operator, and i s relevant here since i n quantum mechanics, s t a t i s t i c a l states and observables are represented by operators. By expressing a l l quantities in terms of operations on phase space, both classical and quantal systems are written i n the same manner so that they are more easily compared. For uniformity in notation, a l l phase space transformations w i l l be referred to as superoperators here, whether classical or quantal. into s t a t i s t i c a l state S' w i l l be abstractly denoted by The previous section has stressed the a b i l i t y to represent any s t a t i s t i c a l state as a phase space function. It follows that the transformation ( 2 . 3 . 1 ) may be written as a phase space transformation A linear transformation $ of s t a t i s t i c a l state S ( 2 . 3 . D (3(r,p|r\p,) f ( r , f p , ) d r , d p \ ( 2 . 3 . 2 ) The kernel (r,p(r',p') of the integral transformation represents in phase space, the superoperator This association "between superoperators and phase space transformations also applies to linear transformations of observables. For example, the mapping from observable A to observable A•, A' = oS A ( 2 . 3 . 3 ) i s r e p r e s e n t e d i n phase space by the k e r n e l p$ ( r , p J r ' , p * ) •V 1, M f of the i n t e g r a l t r a n s f o r m a t i o n ( r . p l r ' . p M a t r ' . p ' J a r ' d p ' . (2.3;*) There are formal differences between state and observable superoperators corresponding to whether the superoperator acts in or &, but these differences do not appear expl i c i t l y in their phase space representations. Obviously the product of successive superoperators i s represented 28. by appropriately composing their integral kernels. Time evolution i s described i n the Schrodinger picture as a linear transformation of states. Since time transformations of a dynamical system form a group, they are equivalent to the exponential of a generator <^ , namely S(t) = exp£-i^(t-t ' f j S(t'). (2.'3.5) This abstract description i s valid for both classical and quantum mechanics. In phase space, the transformation ( 2 . 3 . 5 ) is represented by 0 (p f r.p^ l r'.pWJfCr'.p'Jdr'dp'. ( 2 . 3 . 6 ) The kernel (r,p,tIr'ip'ft *) represents the dynamical displacement superoperator exp£-ii>C(t-t* )J "» On differentia-ting Eq. ( 2 . 3 . 5 ) 1 the time rate of change i ^ S ( t ) / ^ t = c ^ S ( t ) ( 2 . 3 . 7 ) of S(t) i s obtained. This i s represented in phase space by i ^ f ( r f p / t ) / e ) t = < < r , p | i ) s ( t ) / < ) t Z ^ Yj  d£' d- p' = f | ^ ( r , p | r \ p , ) f ( r \ p ' | t ) d r ' d p \ (2.3.8) with the kernel ( r , p j r * f p*) representing©^. In both c l a s s i c a l and quantum mechanics, i s the sum of a free k i n e t i c part % , and a p o t e n t i a l part 1^" . These are now described i n d e t a i l . 30. 2.3.b.CLASSICAL MECHANICS 11 In classical mechanics, Hamilton's equations ^ r ( t ) _ pjO . ^p(t) = )v[rjt)] ( 2 . 3 . 9 ) govern the motion of a phase space point r ( t ) , p(t) as a function of time t, subject to the i n i t i a l condition r ( 0 ) = r, p(0) = p at the zero of time. Liouville's theorem AJ AJ *J *J then states that phase space volume i s conserved and as a corollary, that the distribution function for the dynamical system satisfies f e l ( 5 - £ l t ) = f c i [ r ( t ' - t ) , p ( f - t ) | t J . ( 2 . 3 . 1 0 ) Written as a phase space transformation, Eq. ( 2 . 3 . 6 ) , this implies that the time displacement kernel i s ( ^ c l ( r , p , t j r \ p ' , t ' ) = S(r(t'-t)-r ' )S(p(t'-t)- < S'). ( 2 . 3 . 1 D Because dynamical motion i s reversible, this can also be expressed as = SrS-t^^'ty^l-i't ( 2 . 3 M 2 ) 31-which is closer in concept to the abstract formalism of exp£-i <^c^ t- t ' )j acting on the state r',p*^> i> Equation ( 2 . 3 . 1 1 ) i s , in contrast, the Heisenberg picture, in which the observable | r , p ^ , changes with time. After a l l , Hamilton's equations describe the motion of the observed phase point and the equality of Eqs. (2^3.11) and ( 2 ; 3 . 1 2 ) describes the equivalence between the Heisenberg and Schrodinger pictures. Liouville's equation ^ c l ^ ' S l ^ / f * = - ( P ^ ) ^ f c l ( r , p | t ) / ^ r + ( ^ V ( r ) / ) r ) -if ,(r,p/t)/<)p ( 2 . 3 . 1 3 ) i s obtained by differentiating equation ( 2 ; 3 . 1 0 ) . In this way the evolution generator, Liouville superoperator o^c^» i s represented as a differential operator in phase space. Re-expressed as an integral phase space transformation, Eq. ( 2 . 3 . 8 ) , the the kinetic part the kernel representing 6^cl i s the sum of /^(r,plr\p') = - i ^ ( p - p ' ) ( P A t ) 0£(r-r')/ir (2.3.14) and the potential part ^f/(r,p|r\p') = i S(r-r ' ) ( i V(r)/ir)-) £(p-p' )/i p. ( 2 . 3 . 1 5 ) The classical time displacement kernel of Eqs. ( 2 . 3 . 1 1 ) and 32. ( 2 . 3 . 1 2 ) also s a t i s f i e s the Liouville,-equation ( 2 . 3 . 1 3 ) . That c l a s s i c a l dynamics leads to a unitary group of time displacement operators, when acting upon square integrable phase space functions, was f i r s t noticed "by Koopman . Under these circumstances, St., i s a s e l f - a d j o i n t phase space transformation. 33. 2.3.c.QUANTUM MECHANICS In quantum mechanics, time evolution i s governed by the Schrodinger equation for the wavefunction or, for the operator representative j 2 ( t ) of the s t a t i s t i c a l state S(t), by the quantum Liouville or von Neumann equation i ^ > ( t ) / ^ t = <^ Qy?(t) = *r--[H,/(t)]_. (2.3.16) The integrated form i s J>(X) = expj^ioZ^(t-f)Jy? ( f ) = exp[-iHXt-t^)/&J.^tt ,)expjiH(t ^t , )AJ.* (2.3.1?) This can be rewritten as a phase space transformation, Eq. (2.3.6), of the Wigner function f ^ r . p l t ) , by making use of the associations (2^2.26) to ( 2 ; 2 . 2 9 ) . The integral 14 kernel i s thus ( ? Q ( r , p , t | r ' , p ' , f ) = ^ , p | e x p [ - i o ^ Q ( t - t ^ | r ' , p ^ = h""3Tr A (r,p)exp[-i c£Q( t - t ' ) J£(r' ,p') = h~ 3Tr A(r,p)ex Pr-iH(t-f )Aj A(r\p*) exp j^iH(t-t')/nJ. (2.3.18) 3*. The quantum superoperator O£Q i s self-adjoint as a trans-formation of Hilhert-Schmidt operators, which gets transcribed into a self-adjoint phase space transformation of square integrable phase space functions, just as in classical mechanics. If there is no potential, then the motion i s free and the kernel of the time displacement transformation is which i s identical to the classical result. As a consequence, the quantum Wigner function and the classical state function both satisfy (2.3.19) (2j?3.20) Thus the only difference between classical and quantal free 35. motion depends on what i n i t i a l states are allowed. By an analogous calculation to that in Eq. ( 2 . 3 ; 1 9 ) , the generator °^ quantum free motion is represented by the kernel = - i Stp-p'Kp/V)- ^ S ( r - r ' ) / ^ r = ^ N C I ^ ' P I ^ ' S ^ ( 2 . 3 . 2 1 ) which i s the same as the classical case. The superoperator for quantum interaction i s given in terms of the potential V by if^ f = -n^fy , j>~]_ . ( 2 . 3 . 2 2 ) This can be expressed as a phase space integral kernel in a number of ways. The obviuos direct substitution using the position representation, Eq. ( 2 . 2 . 2 0 ) , for A.(r,p), i s = h" 3Tr A(r,p) l/Q A ( r \ p ' ) • * " V 3 J j ^ r i ^ <5il A C r . p ) | r > [ V ( P 2 ) . v ( E l ) ] . <S 2| A C r ' . p / ) ^ ) 36. = 'n""1(2/h)3 )ex Pr2iR-(p'-p)/fi1 rv(r+R)-V(r-R)l \J L " /W A l •* U * ^ A> A » A J S ( r - r ' ) . (2 . 3 . 2 3 ) dR This form, given by Irving and Zwanzig1^ in 1951, has recently been used 1^ in deriving a modified version 1^ of the phase space representation of the quantum Liouville equation. On recognizing the R integral in Eq. (2 . 3 . 2 3 ) as involving the Fourier transform V(P) = h" 3 dR V(R) exp(iP-BA) = (2.^3.24) of the potential, i t follows that 2^ can also be written as 7 A(r,p|r\p') = ^ i f T 1 S(r-r')v'r2(p'-p)] s m [~2r.(p-p')/n]. ( 2 . 3 . 2 5 ) More commonly, the phase space representation of is written as an i n f i n i t e order differential operator. This is easily obtained from Eq. ( 2 . 3 . 2 3 ) by expanding the potential difference in powers of R, that i s A/ tftelV;^ - 2 ^ 1 ( 2 / h ) 3 S ( r - r ' ) J 0 [ ( 2 n + l ) ! j - 1 jexp[2iR.(p'-p)AJ (R)2n+l 02n +l ( ' ^ / ^ Z n + l ^ _,dR 37. = 2i'h- 1 S(r~r')sin(ih }• • ^ )V(rl) S (p-p«). (2.*3.26) Unlike the free generator, ^Cn» the phase space represen-tation of 2/^ is ex p l i c i t l y dependent on "n and thus not equal to i t s classical counterpart (the harmonic oscillator potential is an exception to this statement). In the limit "h—*0, 2^ becomes equal to 2/^, Eq. (2.3.15). Thus when interactions are present, quantum and classical systems differ both in their dynamics and in the classes of allowed states. 38. 2.4.ASPECTS OF FREE MOTION As discussed in the last section, free motion i s governed by Eq. ( 2 . 3 . 2 0 ) (also Eq. ( 2 . 3 . 1 3 ) with no potential), for both classical and quantum mechanics. This appears somewhat surprising in that wave packet spreading i s always stressed as a quantum effect which (at least by connotation) implies that there i s no classical analogue for this phenomena. When expressed i n a common language, there i s no distinction between the dynamics of classical and quantum free motion; Moreover, free motion maps one phase space point into one phase space point, for both mechanics. The d i f f -erence arises really from what are allowed states. In particular, a phase space point is an allowable classical state, Eq. (2.2.14), which evolves under free motion into the (position) translated point, Eq. ( 2 . 3 . 1 9 ) . Such phase space functions are not allowable states in quantum mechanics. In free motion, the momentum is not changed and thus the momentum distribution also remains unchanged. In contrast, the position d i s t r i -bution (the number density), ( 2 . 4 . 1 ) n 39. i n general, changes i t s shape with time, as well as under-going a displacement. Only i f the momentum distribution i s dispersion free ^  =0, see Eq. (2.4.10)J, does the position distribution retain i t s shape under a uniform translation. Such dispersion free states occur i n classical systems, while the uncertainty principle (2.4.3) limits the allowable dispersions in a quantum system. Thus the shape of the position distribution of a quantum system changes with time, except for the special case i n which "7(1=0 and = °°. Classical systems may also change the shape of their position distributions, or may not, corres-ponding to the value of the dispersion "Tf ? One way of describing the effects of spreads i s by means of moments. At time t, the average and variance of r are = <£>o +/"rl<£>t (2.4.4) and <<5- < s \ <cV>t • <<*- <rV<£- <?>o»>0 40. V^ME-<TV<P - 'P>> + ( P - ^ ^ ^ ^ O ^ O (2.4.5) Since the momentum d i s t r i b u t i o n , g(p) of Eqf (2.4.1), i s unchanged by free motion, moments of the momentum are time Equation (2.4.5) shows e x p l i c i t l y , that i f there i s no spread i n momentum, then the dispersion i n p o s i t i o n i s time independent, a r e f l e c t i o n of the shape independence of the pos i t i o n d i s t r i b u t i o n function n ( r ( t ) i n t h i s case. Otherwise, the po s i t i o n dispersion generally increases with time because the l a s t term i s quadratic i n time, unless the position-momentum c o r r e l a t i o n just happens to decrease the p o s i t i o n dispersion. Again, Eqs. (2.4.4-5-6) are v a l i d f o r both mechanics. The only difference i s that quantum mechanics requires that the uncertainty p r i n c i p l e be s a t i s f i e d . This implies that, i f the dispersion i n pos i t i o n i s f i n i t e , then the momentum d i s t r i b u t i o n must have a spread, and i n consequence, that the pos i t i o n dispersion must change i n time. C l a s s i c a l l y , the spreading i s the same provided the i n i t i a l state i s the same, but there are also c l a s s i c a l states, those of Eq. (2.2.14) type, that do not spread. Supplemented by the equation independent, so no subscript i s written on <^ P> or <pp> . 41. + <^l-<vV(v-<V>f> » (2.4 .6) Eqs. (2.4.4-5-6) describe a l l the time dependences of the f i r s t and second moments of f(r,p). Of particular note, i s that there i s no coupling to moments of higher order. This is exemplified by a Gaussian phase space state function. The prescription of f i r s t and second moments (with vanishing higher order cumulants) i s equivalent to a Gaussian phase space function f(r,p|t) = M ^ ^ x p - f u ^ r - ^ ^ ^ v ^ p - ^ p ) ) 2 -2wt(£-<^>t) , (r<P>n ' (2.4.7) with normalization constant M t ( u t ' v t ' w t ) = 1 T 3 ( u t v t - w t 2 r 3 / 2 - ( 2 . 4 . 8 ) For simplicity, this state function i s taken to be isotropic ( a l l physical directions the same). The average (over spatial directions) standard deviations are then given by ^ r ( t ) 2 = U/3)((r-<r\)Z)t = (-l/3)^lnM t/^u t = I v ^ u ^ - w ^ r 1 ( 2 . 4 . 9 ) 42. and T p ( t ) 2 = (1/3)<(P-<P» 2^ = (-1/3) i l n M t / i v t = ^ ( u ^ - w ^ . 2 ) " 1 , (2.'4.10) while the average eovariance of r and p is ^ r p ( t ) = (l/3)<(r^r> t).(p-< ?»> t = (-l/6)>lnM t/<)w t = -iw^u^-w.,.2)""1. (2.4.11) These equations are easily inverted to express u^, v^ and w^  in terms of the variances, namely u t = * x p 8 ( 7 c r 2 / r p 2 - ^ r p 2 r 1 . and wt - - * ^ . p ( * p 2 /Tp2- A; p 21" 1. (2 rt. 12) For these relations to be valid and for the moments to make 2 2 physical sense (e.'g. X r - ° ) » i * i s necessary that w^  ^u^v^, or equivalently, that * ^ r p 2 ^ K * 7<p2. While ( p ^ and X_ are time independent in free motion, ^ r V , /< and "^T vary with time according to Eqs. (2;4.4-5-6). The tensorial traces of Eqs. (2.4.5) and (2.4;6) give 4 3 . Kv(*)z = * f r z ( o ) + 2 / * - 1 t ^ p ( o ) ^ - 2 t 2 ; < p 2 and (2 .4 . 13) ^ r p ( t ) = ^ r p ( 0 ) + >* l t 7 rp 2-' (2 .4 .14) It is a property of these equations, that the combination ^ r 2 ^p 2- ^.p 2 i s "time independent, so that the normalization constant M is also a constant of the motion. On solving for ut» v^ . and w^ , and inserting these results into Eq. ( 2 . 4 f ? ) , the explicit time dependence of f(r,p/t) i s given by f(r,p/t) = M Q ^ e x p ^ u ^ r - t p ^ - 1 - ^ ) 2 -vo (r<'p> )2-2wo (£- tp/-"1-<£>o )' (r<lp> )I/ = f ( r - ^ t p . p / o ) , (2 .4 . 15) which exp l i c i t l y verifies Eq. (2.3.-20) for this particular example. The correlation ^ L ^ ( t ) between position and momentum rp increases with time, Eq.1 ( 2 . 4 . 1 4 ) , and must be included in the general time dependence. Another aspect of this motion i s brought out by examining the time dependence of the number density n(r|t), Eq. ( 2 . 4 . 2 ) , in the Gaussian case. On performing the momentum integral, n(rit) becomes n(r|t) = [ 2 7 r r / ( t ) ] - 3 / 2 e x p [ ^ (2 .4 . 16) 44. with "^.(t) given as a function of time by Eq. ( 2 . 4 . 1 3 ) . It i s thus seen that the number density i s a Gaussian which translates with velocity <"p^t^, but whose spread increases with time. If at the zero of time, ^ f r ( 0 ) = / ^ p ( 0)=0, then n(rjo) i s a Dirac delta function around ^r^g. The standard deviation now increases linearly with time to describe free d r i f t motion from a point source, for example with a thermal spread in momentum 7^^=ym.kT. Such an i n i t i a l state violates the uncertainty principle, so this must be a purely classical phenomenon. The time dependent Gaussian, Eq. ( 2 . 4 . 1 5 ) , is valid for the free motion of either classical or quantum systems, the only difference being that the uncertainty principle must hold, at a l l times, for the quantum case. In c o l l i s i o n theory, i t is common to consider states with a definite momentum, thus the limit ft —>0 i s required. If a semi-classical description i s desired, the limit ^ -> 0 i s also involved. However the order of the limits is very important. If the n -» 0 limit i s taken f i r s t , then Eq.. ( 2 . 3 . 2 0 ) i s s t i l l valid and there i s no uncertainty constraint, thus classical mechanics i s obtained. On the contrary, i f the /\ —* 0 limit i s taken f i r s t , then , which can be P r non-classical even after taking the limit "n-> 0. The evolution equations ( 2 . 4 . 1 3 ) to ( 2 . 4 . 1 6 ) include the description of the spreading of a Gaussian wavepacket. For a quantum pure state, Eq. ( 2 . 2 , 1 7 ) , the equality in Eq. ( 2 . 2 . 3 0 ) holds. Thus to be a pure state Gaussian Wigner k5 function, Eq. ( 2 . 4 . 7 ) , i s subject to the constraints utvt-„t2=-ft"2 or * T r 2 * p 2 - Xr/= i n 2 . ( 2 . 4 . 1 7 ) The constancy of and of M. mentioned i n r p rp t obtaining Eq. ( 2 . 4 . 1 5 ) i s thus seen as a reqirement that a pure quantum state remains a pure quantum state under free motion. Equation ( 2 . 2 . 1 7 ) for the density operator may be obtained from f(r,p) by Eq. ( 2 . 2 . 2 7 ) t which can be factored in the position representation to give (S'l -H*^ - I"* v t ^ 2 ) - 3 / \ x p [ - | n - 2 v t - 1 ( l + ^ ) ( r ' - ^ : ) ) ; +&-\1£>.r''J , (2.4.18) up to an arbitrary constant phase factor. The time depen-dence, and spreading of the packets follows from Eqs. ( 2 . 4 . 7 ) to (2.4.14). If there i s no correlation between position and momentum at time t, (t) = 0 = w+, then Eq. ( 2 . 4 . 1 7 ) implies that the uncertainty relation, Eq. (2.4.3)» i s an equality. Thus the Gaussian wavefunction with w^=0 is of minimal uncertainty, a coherent state, in quantum optics.' v / 1 lo Prugovecki ' 7 has recently used this as a basis in Hilbert space, for the purpose of defining a phase space represen-tation of quantum mechanics. This i s interpreted here as a fuzzed out Wigner function (this relation i s also given by Prugovecki 3 2) 46. y(R,P ?s) =<R.P;s| j> | R,P.;s> = ( 2/h) 3 JJexp[-(R-r)2/s2-n] expf-(P-p) 2 s 2 / f i ] f(r,p)drdp (2.4.19) A» A, A/ ^ where the Dirac ket |R,Pj^ corresponds to the coherent state identified as ^  i n Eq. (2.4.18), but with the label transcrip-tion ^r)-»R, / P V > P . ^v t-^s 2, wt = 0 (2.4.20) to correspond with Prugovecki's notation. This averaging over phase space i s sufficient to make p (R,P;s) non-negative and thus a distribution function. The Wigner function f(r,p) can be recovered by deconvoluting the Gaussian spreads, so />(R,P;s) i s equivalent to f(r,p) and J A/ A / A> AV to the density o p e r a t o r . W h i l e ^ ( 5 ' P J S ) has the advantage of being a distribution function, the associated observables are changed from their classical expressions (e.g. the kinetic energy) and the time evolution, even for free 8 motion , is much more complex than Eq. (2.3.20) for the Wigner function, due to the"extra" Gaussian spreads^ In view of the equivalence to f(r,p) and the simpler and unique ~ A / (no arbritrary s parameter) properties of f(r,p), i t does A l not seem that the advantage of being positive overweighs the greater complexity of they9(R,P;s) representation, espec-i a l l y since the coherent state representation does not give * 7 . the correct marginal distribution functions . So far, the description of free motion has emphasized moments of r and p and their time dependence. Since the evolution i s generated by the superoperator i t i s useful to understand the time evolution in terms of the eigen-functions (eigenoperators)l(y,*J^> of acting on ^t These are parameterized by a velocity v and wavenumber "7( satisfying the eigenvalue equation with phase space representation = 5(p-/tv)exp(iK'r), ( 2 . 4 . 2 2 ) compare the phase space representation of Eq. ( 2 . 3 . 2 1 ) . The superoperator A can also act on the observable space i n which case the normalization of the eigenoperator i s chosen to be = (ya/2-TT) 3 $(py,*y)exp(i $ ' r). ( 2 . 4 . 2 3 ) 48. These bases are complete in -e/and $ and biorthogonal. = S(v-v') X 1 ) . (2.4.24) This formalism i s valid for both classical and quantum mechanics. The free evolution of a phase space state function can then be written as f ( j , p | t ) % f o , p | e x p ( - i # t ) | s ( 0 ) ^ = f f d v d S <f v,k*'Z>°**l' i-'Z' ? t ) f v , 7 < ( 0 )> (2.4.25) where the expansion coefficients are given by = (A/2TT ) 3 drexpt-i?;. r ) f (r./iv lo). / J ~ ~ * ~ (2.^.26) 2 0 This expansion i s popular in classical s t a t i s t i c a l mechanics 4 9 . when i t is useful to expand the density in terms of wave motion. In quantum mechanics, the phase space eigenfunction [ (v,£ ] ^ is an operator obtained via Eqs. (2 .2 .19) and (2.2.27) to be = drexp(iX« r) A ( r . i i v ) = h3|/iv+!h7^> <^y-inft| . (2.4.27) The kinetic energy s u p e r o p e r a t o r i s now a commutator, namely whose eigenvalue i s (n~* times) the energy difference between the ket and the bra states. More generally, this i s interpreted as the frequency associated with the energy level difference. 50. 2.5.INTERACTING MOTION The general time evolution of phase space state functions has been described in Sec. 3. Central to that description i s the generator of the time dependence, whether classical ^ c i o r quantal One method of evaluating such time dependence^ i s to express the state function f(r,p) i n terms of the eigenfunctions of whether classical or quantal. Since these are rarely known, this i s not a very efficient method and w i l l not be pursued here. Rather, the object of the present discussion i s to explore the properties of the phase space transformation function ^ ( r , p , t l r ' , p ' , t 1 ) , Eqs. (2.3.12) or (2.3.18). It i s expected that a comparison of such classical and quantum functions w i l l point out the similarities and differences between classical and quantum mechanics, with the possibiltity of leading to a greater understanding of semiclassical methods. The detailed form of the phase space transformation; function appears to be on the whole unexplored. In classical mechanics, hamiltonian dynamics always carries one phase point into another, Eq. (2.3.12). The same i s true for quantum free motion and for simple harmonic oscillator motion, since the phase space representation (ry designates that the derivative only acts on the potential V) ^ f ( r , p | t ) / ^ t + (pA)-Jf(r,p\t)/<) r 5 1 . = (2/n)sinf£ i . " . i-1 V(r)f(r,p|t) (2.5.1) |_2 ary <^ p J - -of the quantum Liouville equation reduces in these cases to the classical Liouville equation (2.3.13). This is not to be construed that a phase space point is either a quantum state or observable, such quantities are averages over phase space points with a spread sufficient to satisfy the uncertainty principle. But some quantum dynamics are equivalent to having definite trajectories i n phase space. This i s contrary to the usual perception of quantum mechanics, in which the zwitterbewegung and Feynman path interpretation implies very chaotic, non-deterministic dynamics. However, these statements are made about the time dependence of the wavefunctions, not about the density operator or i t s phase space representation. Whether there are other potentials which have phase point to phase point transformations in quantum mechanics, or what happens i f they do not, are unanswered questions. 14 Leaf has discussed the possibility of evaluating ( ? ( r , p , t l r \ p \ f ) from the Feynman path integral represen-tation of the wavefunctions. It i s within this format that semiclassical approximations have been formulated, see for example Pechukas 2 1. The works of M i l l e r 2 2 and Marcus 2 3 are closely related to this approach. However, no explicit evaluation of (P (r,p,t|r',p*,t') has been undertaken along these lines. Rather than trying to solve the complete time 52. dependence of the phase space function f ( r , p | t ) , i t may be sufficient to calculate (approximately) some moments of f ( r , p l t ) . One way of calculating moments i s to solve the dynamical equations for the moments, that i s , i f A i s an observable, then the moment equation d<A>t/dt = < i ^ A ^ t (2.5.2) is valid for either mechanics. Equation (2.5.2) is not closed in the sense that the moment<^A^ i s not determined by ^ A^.. The obvious procedure is to calculate a hierarchy of dynamical equations, for <£. A, etc. Usually this i s non-ending. It i s then a matter of truncating the sequence at some stage with an ad hoc closure relation. Such a scheme is now described. A procedure of this kind has been advanced by Heller for the study of wave packet motion. Here, a connection i s made to Heller's work and a distinction is drawn between his closure procedure and the one described here. For translational motion, the obvious expectation value to start with, is the centre ^ r ^ of the packet at time t. This changes in time according to the average (drift) velocity d £ > t / d t s<l>t^' ( 2 ' 5 ' 3 ) which in turn has an acceleration due to the average force 53. d<p>t/dt = ^dv ( r ) / i r^ - t , (2.5.4) Equations (2.5.3-4) are valid both classically and quantally, 24 which i s an expression of Ehrenfest's theorem . If the average is over one phase space point, then the classical Hamilton's equations: ( 2 . 3 . 9 ) are obtained. This of course assumes that the state function i s Dirac delta li k e , Eq. ( 2 . 2 . 14), at a l l times, which i s possible classically but not quantally. More generally (classically and always for allowed quantum states), there is a spread in positions (and momenta). If these spreads are small, i t i s reasonable to expand to keep terms only of second order (variance and covariance) in the spreads. In this approximation, the force equation becomes d C P V d t - - ^ « j > t ) / S r t - I ^ 3 V ( < r > t ) / ^ r > t i 3T r(t). ( 2 . 5 . 5 ) involving the variance-covariance position tensor y r ( t ) • <<r-<s>t%-<rv>t < 2 -5-«> and the third order (tensorial) derivative of the potential evaluated at ^ r ^ . . The f i r s t order term vanishes and the higher ones are ignored i n this approximation. It i s now necessary to know how the tensor ^ (t) 5k. varies with time. This i s obtained from Eq. ( 2 . 5 . 2 ) and leads to <)j-r(t)/it v " T f r p ( t ) + & r ( t ) ] ( 2 - 5 ' 7 ) involving the eovariance tensor for the position and momentum, together with i t s tensorial transpose Thus another new moment i s involved. Eq, ( 2 . 5 . 2 ) is used again, to give Ay ^ ' -/A'1 KM - ^ r ( t ) ^ 2 V ( ^ r > t ) / ^ r > t 2 , (2.5.10) involving the momentum variance-covariance tensor and other quantities that have already been defined. The moment equations are closed by applying Eq. ( 2 . 5 . 2 ) to the 55. momentum variance, namely = - r ? p r ( t ) ^ 2 v ( < r V / W ( 2 . 5 . 1 2 ) In these equations, approximations have been made to trun-cate the moment equations at second order in deviations from the mean position and momentum. The truncated moment equations ( 2 . 5 . 3 ) to ( 2 . 5 . 1 2 ) are consistent with angular momentum conservation ( £ i s the Levi-Civita third rank antisymmetric tensor) S/rxp>/<h = - i e X 7 f M < t ) - ; r r ( t ) | / ^ t = o " L ( 2 . 5 . 1 3 ) and energy conservation. To demonstrate the latter, the average energy i s calculated as <H>t =<(g2/2>>+V(r>>t to second order in the spread of position and momentum, where ~f( - U: 7$ i s the tensorial trace of X . The time P % z. P ~ P 56. derivative of ^ H V then follows as + [iv«r>t)Ar>t].)(r>t/ >t +i[)2V«r>t)/^r>t2] • S £r(t>/Ji H j r ( t ) : [ P v ( < r > t ) / J < ^ t / ) t = 0. (2.5.-15) This moment method is valid both classically and quantally; the only difference arising in the possible i n i t i a l conditions that may be used. h. Heller has approached the quantum dynamics of a pure state by a comparable approach. He assumes, for one dimensional translational motion, that the wavefunction i s Gaussian about a time dependent mean position x +, namely It is assumed that x. and p, are real, while the parameters 7*(x,t) = exp [(i/n) * t(x-x t) 2+(i/n)p t(x-x t)+(i/h)Y t ] . ( 2 . 5 . 1 6 ) oC^ and V t are generally complex. Normalization, in fact, requires that (2.-5.17) 57. where oC^. i s written in complex form * t = eit* + ^ t " . ( 2 . 5 . 1 8 ) Various expectation values can then be calculated as o A = <^|PI*> -PT. A x 2 ( t ) =<(x-x t) 2> t. ik«t")-\ and Re * ; p ( t ) = * ( ( x . x t ) ( p - p t ) + ( p . p t ) ( x - x t ) > t = i ,h^ , (-< t "r 1 . ( 2 . 5 . 1 9 ) These are consistent with Eqs. ( 2 . 4 . 7 ) to ( 2 . 4 . 1 8 ) . In making this comparison, the equations of Sec. 4 have been reduced to one dimensional form and the real part,V^. =Re^» of ^  is at this stage arbitrary. The time evolution of the wavefunction is approx-imately described by following the time evolution of the y ' 4 five parameters x^ .t p^, <K|.'»°y' and . Heller assumed that the mean position and momentum follow the classical 58. Hamilton's equation ( 2 . 3 . 9 )» while the time dependence of the complex parameter satisfies On comparison, this i s equivalent to the three moment equations ( 2 . 5 . ? ) , ( 2 . 5 . 1 0 ) and ( 2 . 5 . 1 2 ) , with the Gaussian pure state condition, Eq. ( 2 . 4 . 1 7 ) , automatically satisfied at a l l times. Heller based his procedure on the concept that the potential is locally quadratic, in which case the third derivative of Eq;^  ( 2 . 5 . 5 ) vanishes. As well, Heller emphasizes equating powers of x-x^ in the time dependent Schrodinger equation, rather than moments of x-x^ and of the momentum. It is interesting to note that the modification of the trajectory due to the dispersion in positions, Eq. ( 2 . 5 . 5 ) . involves a third derivative of the potential in a manner exactly analogous to the f i r s t quantum correction to the classical Liouville equation (2.5.20) ^ f ( r , p \ t ) / ^ t = -/T 1p.>f(r,p|t)/^ r +dv/)r)* " " / j " A l A/* A. ^ f ( r , p | t ) / ^ p -(l/24)h 2(^ 3V/^r 3)i^ 3f(r,p)t)/<)p 3 AV M A» ** ~ A. 1 A» • o f t 4 ) , (2.5.21) in the expansion of Eq. ( 2 . 5 . 1 ) 59. The phase of the wavefunction is determined in a moment method, by evaluating dx. As a result of this, H ^ . must satisfy Mt/h « p t *x t/} t -<H>t+ tx2 <) o(x/) t = ( 2 ^ ) " ^ * -V(x t) -^fa^'+yZ^x*'. ( 2 . 5 . 2 2 ) The imaginary part of this equation i s consistent with Eq. ( 2 , 5 . 1 ? ) while the real part determines the time depen-dence of y^i - Eq. (2.-'5.22) i s also obtained by Heller. The total energy ^ H^. i s naturally conserved in the moment method, Eq. ( 2 . 5 . 1 5 ) . while in Heller's method, the third order derivatives of the potential must be consistently ignored i n calculating the time derivative of the total energy. 60. 2.6.THE MILLER SUPEROPERATOR The time evolutions of free and interacting motion have been described in phase space by phase space trans-formation functions. Any difference between these motions is due to the interaction potential, and is responsible for scattering phenomena. Formally, the difference between free and interacting motions gives rise to a Miller super-operator which can be expressed as a phase space trans-: . formation function, both in classical and quantum mechanics. The formalism is illustrated by an explicit calculation for classical mechanics with a hard sphere interaction, appro-priate for the description of a scattering process. Provided the intermolecular potential is sufficiently short ranged, the motion of two interacting particles can be, at long times, asymptotically the same as free particle motion. This is true as long as the interacting particles are not i n a bound state. The free evolution of a state i s described formally by ( 2 . 6.1) while interacting motion satisfies S(t) = exp(-i £ t)S(O). ( 2 . 6 . 2 ) Scattering theory requires finding an interacting state S(t) 61. which converges in the infi n i t e past, to a given free state s(t), lim | | s ( t ) - s(t) t->-co = lira = lim t"»-e»vy j|s(0) - e x p ( i / t ) e x p ( - i / f t ) s ( 0 ) | / t ) e x p ( - i / t t ) | r ,p*| f(r ,,p')dr ,dp' I drdp = 0. (2.6.3) Here the norm for the s t a t i s t i c a l states is equated to the norm of their phase space representatives, which i s equivalent for quantum systems to the trace class norm. The 21 latter was f i r s t discussed by Jauch et a l J for quantum ?6 systems, while the L 2 norm has been used by Hunziker for classical scattering.1* The convergence of Eq. (2.6.3) defines the class of scattering potentials, and at the same time defines the Miller superoperator T = lim exp(i^t)exp(-i^Lt). (2.6.4) A "backwards" superoperator can also be defined using the limit t-»•*-•<> . The formalism i s the same in quantum and classical mechanics although the generator o£ i s different 62. in the two mechanics. There has been quite a lot of dis-cussion2"^ in the past few years about the quantum acting on density operators, while in classical s t a t i s t i c a l mechanics the streaming operators entered as compensating factors in Bogoliubov's classic dicussion of kinetic theory. That these are aspects of the same formalism has recently v 1 been stressed also by Prugovecki . It follows from Eq.v ( 2 . 6 . 3 ) that the interacting state S (0) at the zero of time is related to the free state s(0) by S (0) = SiTs(0), ( 2 . 6 . 5 ) or in the phase space representation According to the intertwining relation ( 2 . 6 . 7 ) i t follows that, i f s (or f) is a stationary free state =0), then S (or F) i s a stationary interacting state (c^S =0). For stationary state scattering, the free motion 63. should correspond to a spatially uniform beam of particles with a definite momentum2^'30 (see also chapter 3 ) . Such a non-normalized state i s f(r,p) = S(p-p'). ( 2 . 6 . 8 ) The associated flux, Eq. ( 2 . 2 . 3 4 ) , i s j- „(r) = v i n e ~ (p^u)f(r,p)dp = p/^a. , ( 2 . 6 . 9 ) having (p'/^0 particles per unit area per unit of time. From this,the stationary state flux of particles around the scattering object i s then j(r) = f(pA)F(r,p)dp = SSWfifa^fc (2- 6- 10' this being determined completely by the phase space trans-formation function representing-U--^. This formalism is valid both for classical and quantal systems. The description of rigid sphere scattering in classical "mechanics i s used in the remainder of this section to exemplify these ideas. In chapter 3 , time dependent scattering i s discussed within the 64. phase space representation of a mechanical system, and 29 connections are made there, to previous operator form-ulations of the co l l i s i o n cross section. In classical mechanics, dynamical motion transforms one phase point into another, E q . (2.3.12). The combination (2.6.4) of free and interacting motion can also be written as This has used the time reversibility of the dynamical motion 65. to introduce r(t) and p(t) in place of r"(-t) and p"(-t), as solutions of Hamilton's equations ( 2 . 3 . 9 ) . The displace-ments A (r,p) = lim rr+p(t)t>:-r(t)T t ~ t->-<*> L ~ / • * •-> and = ^  J|Tp(-(»)-p(r)]dt ( 2 . 6 . 1 2 ) — o o A (r,p) = lim fp-p(tjl = - J ^ v [ s(t)]/^[r(t)J dt ( 2 . 6 . 1 3 ) - o o converge i f the potential dies off sufficiently rapidly at large distances. An interpretation of these displacements i s that, starting at the phase space point r,p, an interacting trajectory is traced backwards in time to a large negative time t when the particles are essentially free, and then forward along a staight line trajectory m^omentum p(t ) J for the same time interval, to arrive at the displaced phase point r- A , p- A . It is also possible to consider obtaining the displacements in the opposite order, i . e. backwards along the free trajectory and forward along the interacting, this actually what the Miller superoperator, Eq. (2.6.4), is describing. For steady state scattering, the classical flux, Eq. (2.6.10), i s thus given by j(r) = f(p/^) STp-P*-A-(r,p<)] dp. (2.6.14) ~J U " / *• A/ P ~ O J M 66. This is now evaluated explicitly for rigi d spheres. Classical dynamics with a rigi d sphere potential i s particularly simple, reducing entirely to a problem in geometry, see Fig. 2 . 1 . Corresponding to an i n i t i a l incoming momentum p', there is an exclusion cylinder C , in position space where the projectile cannot enter (by classical mechanics). For i n i t i a l positions r* oirtside this exclusion, that i s , r' fi^C ,, then no c o l l i s i o n occurs a n d - f l ^ u is ~> ^ p .L-.hs the identity, whereas for r* within the exclusion cylinder, the phase point r',p' i s changed by a collision, see Fig. 2."1( On combining these results, the phase space representation is o f i l c l L,hs + S(p-p'+2uu'p') S(r-r'+2*uu-p')C . t r ' ) . ( 2 . 6 . 1 5 ) ~- ~ A/ -v v P ~ A The unit vector u and distance X are designated in Fig. 2 . 1 , and given by and u = a " 1 ( r I - ( 2 . 6 . 1 6 ) A/ rf-r*. p* + r a 2+(r'. p ' ) 2 - r ' 2 ] * ( 2 . 6 . 1 ? ) in terms of the hard sphere radius a and the i n i t i a l phase point coordinates r',p*. In Eq. ( 2 . 6 . 1 5 ) , the symbol C ,(r') Ay P <v 67. Fig. 2.1.Geometry of a Classical Hard Sphere Collision 68. i s used to denote the characteristic function for the exclusion cylinder C ,. The classical hard sphere phase space state function F 0} , (r,p), Eq; ( 2 . 6 . 6 ) , which describes the interacting p , ns " *s motion corresponding to the i n i t i a l stationary state ( 2 . 6 . 8 ) i s given by dr' S(p-p')["l-C ,(r ' ) l + fdr' £ ( p - p'+2uG . p ' ) S(r-r'+2 * u G . p')C.(r'). (2.6.18) Since and u are functions of r' (andp'), i t i s incorrect to evaluate the r' integral by considering the position delta function to be of the form £(...-r') f rather i t has the functional form S(r-R), where / l A A . R = r'-2^uu-p' ( 2 . 6 . 1 9 ) is a function of r' and p*. The r* integral is appropriately ~ ~ /j evaluated by a change of variables, from r' to R, with the Jacobian dr' dR i -a^p'' u (R2-«2)(2<X-.ap'. u) (2.6.20) This leads to the formula 69. p',hs vJ (r.p) = r> a3p^«u £(p-p'+2uu-p') V P - P ' ) - — 7 — 7 — 7T~ (r 2-« 2)(2* - ap*'. u) L - A / ( 2 . 6 . 2 1 ) where the requirement r'£Cp, i s identified with the requirement that r 4. C ,. The vector R is now identical to ' p r, due to the delta function, and r'(p',r) i s obtained by inverting Eq. (2.6.19). At the same time, the unit vector u and distance are to be considered as functions of p' and r. A E x p l i c i t l y , u is given by u = (r-*p' )| r-«p'| -1 (2.6.22) while oC is the solution of a r 1 A / 2 ,2 = r -o\ . (2.6.23) From Eq. (2.6.21), i t i s seen that the distribution function is the sum of two terms, one due to free motion and one due to scattering. This i s indicated in Fig. 2.2, which demonstra-tes the individual trajectories. Finally, the particle flux at positioner, Eq,(2.6.10) is given by P' " 3 / S. / 1 „ A-A- . a^p'' u( p -2uu»p') A / W ( r 2 - * 2 ) ( 2 * -ap'-u) (2.6.24) 70. 71. where again u and oC are functions of r and p', Eq.: (2.6 .22) and (2.'6.23). The flux lines, or streamlines, are given in Fig. 2 . 3 . This demonstrates how free and reflected motion combine to depict f l u i d flow around a semi-infinite sphero-cylinder. In fact, this i s more than an analogy to f l u i d motion. Equation (2.6.24) is precisely the stream velocity at position r, corresponding to incompressible inviscid f l u i d flow around a spherocylinder. That i s , the average velocity j(r) satisfies the steady state continuity equation ( V^r).j(r) = 0 (2/6.25) together with the boundary condition A n«j(r) = 0 ( 2 . 6 . 2 6 ) surface of zero normal velocity on the surface of the sphero-cylinder. The f i r s t of these equations can also be obtained directly from the time independent form of Liouville's equation (2.3.13)» where the potential term also vanishes outside the sphere. On the surface of the sphere, the r i g i d sphere force i s i n f i n i t e and directed normal to the sphere. Thus, by the Liouville equation (2.3.13). the normal derivative n«^f/)p must vanish on the surface of the r i g i d sphere. In this way, the boundary condition n-jjr) r = a = J n - ( p ^ ) f | r = a d p = i [ ( p 2 / ^ ) ( n ^ f / ) p ) \ r = a d p =0 73. ( 2 . 6 . 2 7 ) can "be directly obtained from the dynamical equation. That there is no flux inside the cylindrical part is due to the boundary conditions j = p'/n. imposed on the f l u i d flow at r.p* = -DO? Ay ^ The generalized differential cross section is related to the flux at large distances by (see chapter 3) ^ g s c < J > r ) - l i m r 2 r A - ( J ( r ) - j i n o ) / 5 l n o ( 2 . 6 . 2 8 ) where the incoming flux j j _ n c is here p*/^ . On evaluating the large r dependence of the hard sphere flux, Eq, (2.6.24), the generalized differential cross section is &Jv'-*$) = *a 2 - lim C .(r)fr 2r.p'+ia 2 7. ( 2 . 6 . 2 9 ) S s c ~ r _ » o o P ~ L J As r-*e», the requirement that r£ C , implies that r and p' have nearly the same direction. The singular term in Eq. ( 2 . 6 . 2 9 ) can be integrated over f d r lim C . ( r j p r . p ' + i a 2 ] = lim fdr C ,(r) f ^ r - J ' + i a 2 ] r-o, J P - L J = lim 2ir ( 1 1 dcose- r 2cos 6 +|a2 r-»oo J ( l - a 2 / r 2 F i -* 74. = lim £r 2ri-(l-a 2/r 2)1 + a 2 f l - ( l - a 2 / r 2 ) * ] f 2 . (2.^6.30) I T a . In consequence, the generalized cross section i s explicitly-obtained as CT (p'-5>r) = i a 2 --rra2 £ ( r - p " ), ( 2 . 6 . 3 1 ) gsc *j which i s the standard result. The quantum analogue to Eq. (2.6.24) and the classical plot, Fig. 2.3, i s the streamline calculation of Hirschfelder and Tang^ . There the streamlines show interference effects, see Fig. 3 of their paper. Also present are dispersive effects which partially illuminate the cylindrical shadow region. CHAPTER 3 SCATTERING AS A TIME DEPENDENT PROCESS 76. 3.1.INTRODUCTION In textbook discussions of scattering , i t i s normal to describe classical scattering in terms of time dependent trajectories, which are then related to cross sections. The quantities used in this description are observables and the i n i t i a l state i s usually a classical pure s t a t i s t i c a l state (see chapter 2 ) . On the other hand, quantum scattering usually involves the calculation of stationary state Dirac wavefunctions, while the observables enter only in making a physical interpretation of the result. Taylor's 3 treatment is a counter-example to this general statement, but even there, the emphasis i s to derive the S matrix formula for the cross section, rather than to formulate scattering theory in a manner in which time dependent concepts are retained and possibly used for approximation procedures.' What i s striking, i s that not only are the conceptual pictures of the scattering event different for both mechanics, but so are the mathematical languages used to describe the scattering event I It thus seems that, while semi-classical approximations to quantum scattering may end up with formulas that look like their classical counterparts, the actual formalism and approximation methods have very l i t t l e input from classical mechanics. Rather the semi-classical methods seem to be mathematical expansions in powers of "n with l i t t l e or no associated physics. As seen here, the major reason for this state of affairs, is that classical 77. and quantum scattering are expressed in very different ways. It is the purpose of this chapter to not only formulate classical and quantum scattering in both conceptual pictures, namely stationary state beam and time dependent trajectory approaches, but also to express them in a common mathe-matical language. In this way, i t i s hoped that a better understanding w i l l be obtained, of what i s classical versus what i s quantum, i n a particular scattering process, with the ultimate aim of providing better approximations to the internal workings of the col l i s i o n dynamics. The remainder of the introduction comments on the common mathematical language to be used for the description of the scattering event. The square root of the classical distribution has been compared with the coherent state representation of the wavefunction , a formalism which i s viewed as a r t i f i c i a l here. It has the advantage however, that in both classical and quantum mechanics, the absolute square i s a distribution function^. But i n the quantum case, the position and momentum labels cannot be interpreted as the position and/or momentum I k of the particle . In the present treatment, what is stressed, is a common language of observables and s t a t i s t i c a l (in contrast to Dirac wavefunctions) states. Classically, these quantities are expressed as phase space functions, while in quantum mechanics, they are expressed as operators. 78. But expressed in this way, the formal structures of classical and quantum mechanics are much the same. Moreover, i t is also possible to write quantum mechanical expressions as phase space functions (only translational motions are discussed i n this chapter).This phase space representation of quantum mechanics allows the classical and quantum scattering processes to be described in a common language. Then only the details differ i n the two dynamics, but those are exactly the effects about which a better understanding i s desired; In both classical and quantum mechanics, the physical observables are considered to be elements of an algebra. S t a t i s t i c a l states are linear functionals on this algebra. If one goes a step further, i t i s recognized that the states are elements of a vector space and the physical observables are particular linear functionals on this vector space. Since the s t a t i s t i c a l states are usually considered to contain the time dependence (the Schrodinger picture), then i t is this latter form which i s most convenient. The expectation value of a physical observable A in the s t a t i s t i c a l state S w i l l then be the scalar product of A and S, to be written in the form (see chapter 2) (3.1 . D Although physical quantities are real, this is formally 79. linear i n the s t a t i s t i c a l state S as an element of the state of the observable space compare the discussion in chapter 2. The distinction between the observable and s t a t i s t i c a l state space is particularly important i n classical mechanics (see chapter 2). While the description given above is very abstract, these quantities are usually expressed as phase space func-tions in classical mechanics and as operators in quantum mechanics. To further the unity of presentation, i t is stressed that the quantum mechanical operators can be represented as phase space functions, the Wigner equivalent representation, so that Eq. (3.1.1) can always be expressed exp l i c i t l y as an integral over phase space (only trans-lational motion is considered in this chapter). As obser-vables of physical interest, for example the particle flux, have a classical as well as quantal meaning, the phase space function which represents the quantum observable is (usually) identical to the classical observable. On the other hand, the phase space representation of a general quantum s t a t i s t i -cal state, the Wigner function, i s not a distribution function, whereas the phase space representation of a class-i c a l state is always a distribution function. Thus, in general, the Wigner function cannot be interpreted in a classical manner even though a l l quantum expectation values antilinear in the observable A as an element 80. may be calculated from i t . Conversely,there are classical states which have no quantum interpretation. Primarily, the latter are those which violate the uncertainty principle. It i s thus inherent in the present treatment, that spreads in the position and momentum be allowed in the description of the c o l l i s i o n events, since these are required in the quantum case. The c o l l i s i o n process i s formulated completely within a time dependent viewpoint. Time dependent trajectories may be described for both classical and quantum mechanics 7 using Ehrenfest's theorem. Recently, Heller' has described the possibility of approximating quantum dynamics by trajectory equations in a manner similar to the moment method proposed in chapter 2. Thus quantum scattering may be formulated from the same conceptual picture - in terms of time dependent trajectories - as classical scattering usually i s . On the other hand, i f the spread in position is allowed to approach i n f i n i t y , the quantum mechanical stationary state beam picture results, which i s also valid i f the mechanics i s classical. In this way, both trajectory and beam pictures of scattering may be applied to either mechanics. Special choices of i n i t i a l conditions involving the spreads and choice of breakup of "forward" scattering leads to the "ordinary" differential cross section and to 8 9 previously described , 7 generalized cross sections. 81. 3.'2.THE COLLISION PROCESS A number of particles, i n a particular s t a t i s t i c a l state, are directed at a target. After colliding, they w i l l leave the target i n different directions. The cross section deals with the probable number of particles coming out i n a particular direction per unit flux of incoming particles. This i s consistent with the beam picture of a molecular beam experiment ,whe re by apertures define what particles enter and exit the scattering chamber. It i s also consistent with the picture of kinetic theory c o l l i s i o n processes, i n which the incoming momentum has a dispersion 7Yp equal to k T ^ being the reduced mass, T the temperature while k i s Boltzmann's constant). Moreover, i t also gives a s t a t i s t i c a l picture for a single co l l i s i o n event, which i s the time dependent trajec-tory picture. The description i s thus quite general and applicable to both classical and quantum mechanics. In fact, the general expressions developed for the cross section are valid for both classical and quantum mechanics. The differences arise only when the detailed dynamics of the c o l l i s i o n event are considered (this being outside the scope of this chapter). If the particles moving towards the target are far from the target, then their motion should be free. This motion implies that there w i l l be an average momentum < P > - / p | s < t j | • If (3.2,!) 82. which i s time (t) independent, and an average position (Sfl*> " <^£ ! s ( t % = 5 + ( p " / ^ ) ( t - t 0 ) . 0.2.-2) corresponding to linear motion. The average impact parameter b i s perpendicular to p" (b»p" = 0) and the time t~, which w i l l henceforth'be taken as zero, is the time when the center of the distribution i s closest to the target, on the basis that there is no interaction between the target and particles. ju. i s the reduced mass. This description is valid both class-i c a l l y and quantally. For the purpose of counting which particles collide, i t is required that the particles must move through a plane perpendicular to the average momentum p" placed at some dis-tance TQ away from the target, with the stipulation that is large compared to the range of the potential. For inf i n i t e ranged potentials, TQ must be taken infinite i n a limiting process, but this problem i s not considered here. The center of the packet w i l l thus pass this plane at time t" = ^M.r 0/p". Movement through an area i s governed by the particle flux j(r|t) = (p/tOf(r,plt)dp, (3.2.3) defined using the phase space representation for both class-i c a l and quantum mechanics. Here f is the classical distribu-tion function or for quantum mechanics, the Wigner function representing the density operator. 83. The particle flux at a point r' = b' - r o ? " i n the incoming plane is the value of j ( r l t ) at this point. This AI ~ 1 w i l l vary with time and impact parameter b', see Fig. 3 f l for the geometry of the coll i s i o n process (p" is the unit vector in the p" direction). To calculate the total number of particles that enter, i t i s appropriate to average over a l l impact parameters b' and times t during which the particle can go through the plane. This average w i l l necessarily be in the direction p" although individual particles may travel in Ay different directions. Thus the number of incoming particles i s ,+co N. = p' mc * '-OO " "• " d t Jd<2Vjino<V-'V"U> J - A ~ \J " n f(r,p 0)drdp. ( 3 . 2 . 4 ) ft AJ ' At At This i s identical to the total number of particles in the system, as given by the normalization of the s t a t i s t i c a l state f. In particular situations the particles may be spread over a wide range of impact parameters, or may be quite local-ized around the average b. As well, the time spread might be large or small, these things being determined by the behaviour of the phase space state function f ( r , p l t ) . The general formalism w i l l require N. „ to be f i n i t e , while the- usual mc associations of a coll i s i o n cross section involving a uniform beam of particles i s obtained by a limiting procedure, see Sec. 3.3» Pig. 3.1.Trajectory for the Center of the Phase Space Packet: a i s the range of the potential co 85. After colliding, the scattered particles are moving away from the target. The experimental setup i s to place a counter, or aperture for which every particle that goes through the aperture is counted, at a large distance from the target. Again, the idea of the large distance i s that the particles are again free from the target. If the aperture is placed at a distance R from the target, with the elemental 2 A area R dR (normal to the target direction), then a certain particle flux w i l l be observed, usually by counting the total number of particles going through the aperture. For the incoming pulse of particles N ^ n c entering into the scattering chamber, the total number exiting through the A. aperture at dR i s dN t Q t = R 2dR J R«jlRR|t)dt. ( 3 . 2 ; 5 ) The flux j(Rlt) w i l l i n general not be in the direction /v " A R, even at large distances R. One way of classifying the directional behaviour of j , i s to resolve i t into components in the R and p" directions, namely i ( R t t } = J S c ( 5 l t ) R + ^ e t C R l t J p " . ( 3 . 2 . 6 ) Equation ( 3 . 2 . 6 ) represents a particular ^choice for the resolution of the flux. It i s by no means unique, but is taken here to define the scattered part of the flux. This definition i s applicable whether or not there i s any spread 86. in momentum directions. If " ^ i s the experimental angle of deflection, (0^ & < 7T ,cos^- = fi*p"), then simple vector analysis implies that J 0 o = (R'O - P" 4J oos&) csc2-e-. (3*2 .7) S C *j /v This shows how, in principle by t i l t i n g the observation aperture, one could sort out what i s the scattered flux. The differential number of scattered particles i s then (number per steradian) / ^ 2 dN /dR = R sc o i (R|t)dt. ( 3 . 2 . 8 ) sc - CO The ratio dN /N. dR i s then related to the differential sc mc cress section, see Sec 3 . 3 . Equation (322,6) resolves j into scattered and non-scattered parts. However, this i s experimentally d i f f i c u l t . Rather, what i s done, is to avoid allowing any of the incom-ing beam to reach the observation aperture. Then J n e^. is zero and ,i = R*j. This can be done for non-forward scat-sc X tering by collimating the incoming beam, but not for forward scattering. A description which avoids this d i f f i c u l t y , is to consider the generalized scattering flux Jgao(S|*> " i(B|*> - i i n o ( B l t ) - (3-2-9> .This i s an alternate resolution of the flux j(R|t). Here, 87. the incoming flux i s governed by the equations of free motion and measures that part of the incoming flux which reaches the observation aperture undiminished by scattering. It can be measured by just removing the target. In terms of the scattered flux, the generalized scattered flux i s jgsc = j s c R - ( j i n c - W ^ ' (3.2.10) Physically the non-scattered part of this flux ( j . - j .p" ) \2inc dnet^ is the loss from the incoming beam (at impact parameter [R 2-|p-'R |^J*) due to collisions. Thus the generalized scattered flux describes both the gain and loss terms in a col l i s i o n process. The corresponding differential rate (number per steradian) 00 dNgso/dR = R2 n A — 00 R ' i g s c ( R l t ) d t (3*2.11) is again measurable and includes both gain and loss terms. . A . O The ratio dN /N. dR i s related to a previously defined generalized differential cross section, see Sec. 3.4, 88. 3t3.GROSS SECTIONS The differential cross section w i l l be defined in two different ways, depending upon which conceptual picture i s used - the beam or time dependent trajectory picture. In the beam picture, the incoming phase space function f ( r . p l t ) must represent a uniform beam, while in the trajectory pic-ture, f(r,p|t) i s a moving pulse ior time dependent "packet". The latter picture i s consistent with Heller's approach''' and the moment method given in chapter 2. In application, Heller appears to use his method only to calculate the S matrix, but i t would seem more appropriate to calculate cross sections directly, for example by the methods presented herein. A further difference between Heller's approach and the present one, i s that Heller assumes the center of the packet follows classical dynamics, whereas i n the moment method, the motion of the center i s modulated by the spread of the packet, see chapter 2-& The trajectory method i s the usual classical approach and thus seems the natural one to use to understand semi-classical dynamics. A cross section i s defined as the rate (number per second per steradian) at which particles exit through an angular aperture, i f the incoming flux (number second per cm ) i s uniform and equal to one. One way to satisfy this required uniformity, i s to integrate over a l l average impact parameters b of the i n i t i a l packet, see the discussion following Eq. (3;< 2ih), Thus the differential cross section is the limit 89. C(p"-*R) = lim d ( 2 ) b dN (3^3.1) R->« U N. dR m c in which the dispersion ftp= ^ P - P " > 2 ^ - I (p-p") 2 f^'p|°) dr dp (3^3.2) in the i n i t i a l momentum is required to vanish in order that the c o l l i s i o n process has a well defined momentum. In an analogous manner, the generalized differential cross section is obtained as R-»ro ^ N. dR m c Since equations (3»3»1) and (3*3*3) are appropriate for class-i c a l scattering, then, for that case, i t i s also possible to consider a limit i n which the dispersion i n position would also vanish." This gives the usual classical cross section, 7 as shown i n Appendix A. Heller's method' as well as the moment method of chapter 2 may be directly applied, for quan-tum systems, to the evaluation of Eqs. (3.3.1) and ( 3 . 3 . 3 ) . An alternate method of ensuring uniformity i s to consider an i n i t i a l state function f ( r , p ( 0 ) which has a broad range of impact parameters, thus giving the beam picture. For definiteness, i t i s assumed that the i n i t i a l state function is a Gaussian i n position and momentum, 90. ( 3 . 3 . 4 ) Again, b i s perpendicular to the average momentum p" and at time t = 0, the average position i s b, which i s the closest that the mean position w i l l come to the target. As time pro-gresses, f(r,p|t) i s given by Eq. ( 2 . 4 . 1 5 ) . On the plane £-T>" = - T Q before the co l l i s i o n has occurred, the flux can "be calculated to be / [ l + X ^ t ^ ^ ^ i r ^ Z + t 2 / C ^ 2 ) ] 3 / 2 J exp^(b'-b - r 0 p " - p " t ^ ) 2 / 2 ( i ( 2 + t 2 / C 2 ^ 2 ) ] ( 3 * 3 . 5 ) compare Eq. ( 2 i a 4 . l 6 ) for the number density." If the disper-sion i n position i s very large (in the limit X ~* 0 0 ) while everything else remains f i n i t e (including time t, distance r Q and dispersion # ) , then j becomes uniform , W S ' ^ | * ) j « r ^ ( p ^ K z r t K 2 , ) - ? / 2 , ( 3 ; 3 . 6 ) but formally zero. This i s interpreted as a uniform beam, valid for both classical and quantum mechanics. For suffi c -iently large times and impact parameters, Eq. ( 3 . 3 . 6 ) is not valid. But for large enough % , a l l those impact parameters for which collisions occur, are within the uniform 9 1 . planar distibution. The momentum dispersionK ^ can be taken in the limit to zero, so that beam scattering with a definite momentum is being described. This can also be done quantum mechanically, as long as Kr i s already i n f i n i t e , so that the uncertainty principle i s not violated. In the limit Ar"*<*> f the scattered flux j a_(R|t) becomes time independent. The cross section i s then the ratio of outgoing to incoming fluxes, namely <5"(p"-*R) = lim lim R"-j a e(R\t)/p".j. (R/t) lim lira i2**'V/'{M/va)ir^AR\li)$ R-*» (3.3.7) Note that the Xv limit should be taken f i r s t . In an analo-gous manner, the generalized cross section i s given by R-»« (3.3.8) It w i l l be shown i n the following section, how the two expres-sions for the generalized cross section, Eq. (3.3.3) and (3*3.8), both reduce to the form obtained by stationary state scattering theory.* Formulas for the ordinary cross sec-tion follow in a similar manner. It should also be noted that in taking the limit of ii to zero; to obtain 'the':classical result from the quantum, the order in which the limits are taken i s important. The i i limit i s to be taken before 1\ ^  Off 92. 3.4.SUPEROPERATOR FORMULAS FOR THE CROSS SECTION In the previous section, Eqs. (3.3.3) and (3.3.8) express the generalized cross section i n terms of time depen-dent fluxes;' Here the dynamics of the c o l l i d i n g system i s used to express the fluxes and the cross sections i n terms of c o l l i s i o n superoperators^ I t i s stressed again, that the formalism i s v a l i d f o r "both c l a s s i c a l and quantum mechanics. Central to the c o l l i s i o n problem, i s the r e l a t i o n between free and i n t e r a c t i n g motion. As described i n chapter 2, the phase space state function F(r,p|t) describing the int e r a c t i n g system i s related to the state function f ( r , p j t ) appropriate f o r free motion by the M i l l e r superoperator-H.-^. Expressed as a phase space transformation, t h i s i s (see Eq. (2.6.6)) P ( ^ l t J = j f ^ ' P l ^ L Ir-P'^f f(r'.pMt)dF'dp' J J " N * (3.4.1) The cross sections involve the p a r t i c l e f l u x . For incoming p a r t i c l e s , t h i s involves the free function f ( r , p ) t ) and i s given e x p l i c i t l y by E q ( 3 . 2 . 3 ) . On the other hand,' the t o t a l f l u x f o r the i n t e r a c t i n g motion requires F ( r , p ( t ) , or using the M i l l e r superoperator, j ( r | t ) = |(pA)P(r,p|t)dp 9 3 . If some knowledge o f i \ - ^  is available, as for example for classical r i g i d spheres as discussed in chapter 2, then Eq.a ( 3 . 4 . 2 ) can give a picture of what flow i s like i n a c o l l i s i o n . The generalized scattering flux of Eq. (3.2."9) has a similar form, namely fir'.p'ltjdr'dp'dp. A / \ ( 3 . 4 . 3 ) It i s noticed that the time enters only in f. To calculate the differential rate dN /dR and cross section, Eq1? (3 ."3 .3) , gsc i t i s necessary to carry out integrals over b and t, and also to evaluate N. Since the limit K~* 0 w i l l be taken, i t mc p i s sufficiently general enough to assume f(r,p | 0 ) is Gaus-sian, Eq. ( 3 . 2 . 4 ) . N. is immediately evaluated, Eq. ( 3 . 2 . 4 ) , JLX \ \ * to be 1.' Since b and t only appear in f ( r , p l t ) , these can immediately be integrated over, to give J d t j d ( 2 ) b f ( r , p | t ) = y^/p. Ap " ) ( 2 i t K 2 ) " 3 / 2 - oo e x p [ - ( r p " ) 2 / 2 A 2 ] . ( 3 . 4 . 4 ) It follows that the generalized cross section calculated via Eq. (3 .3»3) i s equal to On L CT (p"-^R) = lim R gen £ f < _,a P - o ( R - P / U L ) < f R , p | l L T - l r'.P' 94. (u/v- P" ) (2ir ft 2 ) " 3 / 2 e x p [- (pip" ) Z/Z % 21 dp • dr • dp / A - P L A * " P J / W " A y = lim R2 ( Vp" )XM f l^1^ ,'S"^ds,d5' ( 3 . 4 . 5 ) which uses the identification that the ^ p "* 0 limit of a Gaussian i s a Dirac delta function. The integral over r' A J implies that the cross section is just a properly normalized flux at large distances for a uniform i n i t i a l beam with momentum p". A y The same result i s also•obtained from the alternate equation for the generalized cross section, Eq. ( 3 . 3 . 8 ) . This requires the limit # 00 to be taken, followed by the limit A^p—> 0 . Since the spreads appear only in the free state function f ( r ' . p ' l t ) in Eq. ( 3 . 4 . 3 ) , then in the limit 'fy # 5 . (2rK p)-3/ 2exp[-( r-b-p-t^)/2X 2 -(p ,-p") 2/2 * 2 1 A^ ' /V P J = lim (2 7 r K 2 ) " 3 / 2 e x P r - ( p ' - p " ) 2 / 2 X 2 I V i - y W A , " P J = S(p/-p-). ( 3 . 4 . 6 ) That this limit can be applied in the present case, depends 95. on the fact that S i r differs from the identity 1 only due to the local intermolecular potential, so that only small values of l r ' | contribute to the integral in Eq. ( 3 . 4 . 3 ) . On combining Eqs. (3 . 3 * 8 ) , ( 3 . 4 . 3 ) and ( 3 . 4 . 6 ) ,the generalized cross section i s again found to have the form of Eq. (3»4.5)» last line. This shows that both Eqs. ( 3 . 3 . 3 ) and ( 3 . 3 . 8 ) yield the same result, a formula for ^ g e n which is valid for classical and quantum mechanics. The generalized differential cross section can be further reduced so that a l l c o l l i s i o n information i s con-tained in the transition superoperator . This may be accomplished by writing the Miller superoperator as a time integral -Q.L = 1 - i j ds exp(i<fs) ^ e x p ( - i ^ s ) , ( 3 . 4 . ? ) using the identity s exp(i«^s) = exp(i/^fs) + i Jdt expd^t) if exp^-i^(t-s)] , ( 3 . 4 . 8 ) and manipulating the time integrals, to give o SLL = 1 - i j d s e x p ( i ^ ) ^ e x p ( - i ^ s ) . • ( 3 . 4 . 9 ) - c o The generalized differential cross section, Eq. ( 3 . 4 . 5 ) , 96. now becomes o C T (p"-»R) = lim (-i)R 2 ds dr'dp (R.p/p") X R + P ^ P [ ^ k,+P"s^'P"^ • ( 3 . 4 . 1 0 ) Various s i m p l i f i c a t i o n s can now be performed. On using the eigenfunctions of see chapter 2 , the state representative becomes d £ d I \^'^ ( ^ / 2 T T ) 3 (^p-^ vjjdr' exp[-iX' (r;+p"s^)J = Ja£ exp(-iX'P"«^ >I<P>.*J^  ' (3 .4 . 11 ) which i s independent of the time s. I t i s now convenient to transform the time (s) i n t e g r a l i n Eq. ( 3 . 4 . 1 0 ) . Define z = -ps^U, so that the i n t e g r a l over s becomes 00 id s ? ^ * £ s ^ ^ l =/\fdz P ^ " Z ^ ? = / A . dz z" 2p n z - p ) U"-z,pJ . (3 .^.12) 97. Here the vector z = zz has also been introduced. Finally, define x = R-z, so that the limit of the observable repre-sentative becomes im R2 ds dp p UR+ps/M,p -*oo J - ~ ^\\- ~ / * lR = lim u. R dxdp p(R-x'"*2 A / <— A« A . S['p-(R-x)| ?-x\- 1M,pl * (dp p2 Jdx ^ » P R | « = R Jdp (3.^.13) The details of this calculation are similar to a previous g calculation of the free motion resolvent. Expressed i n terms of the eigenfunctions of % , this i s oo //. dp p' r dx^(x,pR( = ( 2 T T ) ^ dv v 2 ^(vR,0 ) l . £ X (3.4.14) In consequence, the generalized cross section i s expressed i n terms of the observable and state eigenfunctions of f\, as oo <y (p"-» R ) vgen vit = ^Py^/p") J d v v2<^(vR,0)(-i3| ( P " / > . 0 ^ (3.4.15) This equation describes 0*1 as the \j matrix element between ^ gen an i n i t i a l state of the system with definite momentum p", but which i s spatially uniform, and the observable which selects A whether the particle has a f i n a l velocity direction R with arbitrary position vector and arbritrary momentum magnitude. The entire calculation and result i s valid for both mechanics^ 98. A result similar to Eq. ( 3 . 4 . 1 5 ) has "been obtained by • 11 Prugovecki in his derivation of the #oltzmann equation. His result, differs, however, by a factor of p/p" which i s of no computational consequence for the elastic scattering that i s discussed both here and by him. His results are also valid for both mechanics. 99. 3.6.DISCUSSION Time dependent phase space packets have been used to describe c o l l i s i o n processes. With appropriate use, both the time dependent trajectory and beam pictures have been des-cribed. Since phase space packets have been used, the math-ematics i s valid for both classical and quantum mechanics. It i s hoped that this approach w i l l contribute to a better understanding of the similarities and differences between the two mechanics, as they apply to the c o l l i s i o n process. Interest has recently increased in devising a quantum version of the classical time dependent trajectory method, due to the work of Heller . While Heller has used the time evolution of a quantum wavepacket to compute an S matrix, i t would seem more appropriate to calculate a cross section directly from the dynamical results, say via Eqs. (3*3.1) or ( 3 ' 3 . 3 ) . A method of great similarity, the moment method described i n chapter 2, can also be used to solve the class-i c a l distribution function or the quantum Wigner function.* These methods are appropriate for making semi-classical approximations to the quantum results. To obtain the cross section, i t i s necessary to integrate over a l l impact param-eters b. If the i n i t i a l l y spread packet is Gaussian in momen-tum, i t i s found that the component of momentum along the average incoming direction determines the incoming flux, see the f i r s t form of Eq. ( 3 . 4 . 5 ) . This procedure, averaging over a l l b t has been considered by Taylor , using wavepackets for quantum mechanics. He has obtained the same momentum component. 100. The beam picture corresponds more closely to the sec-ond form of Eq. ( 3 . 4 . 5 ) . This is viewed as being related to the streamlines recently stressed by Hirschfelder and cowor-kers* 0. Such a description could also give a picturesque view of both classical and quantum scattering, together with an elucidation of their differences. The f i n a l form, Eq. ( 3 . 4 . 1 5 ) , for the cross section i s valid for both classical and quantum scattering, involving matrix elements of in i t s phase space (or eigenfunction of *K ) representation. While the present derivation i s from a time dependent approach, an analogous quantum version has 8 previously been obtained from a stationary state approach. That the results are identical w i l l now be demonstrated. For quantum mechanics, the eigenfunctions of ^ .may be written in the form of ket-bra Dirac momentum states ( 2 . 4 . 2 7 ) , namely ( 3 . 5 . D and The difference in factors arising from the dissimilarity in observable and state spaces. In terms of the Dirac momentum states, the generalized cross section, Eq. ( 3 . 4 . 1 5 ) i s oO 0 ^ e n ( p " - > R ) = (^V/p" ) ^ d v ^ = t r | R X H ( ( - 1 )3 (|p">:(h^/p,,)6i ). ( 3 . 5 . 3 ) 101. This i s the same as Eq. (6 3 ) of reference 8. The replace-ment of the scal a r product ^S" \ ^ by the quantum trace t r has been made. The c l a s s i c a l l i m i t of t h i s quantum formula has also been e v a l u a t e d 1 1 . Although only the purely t r a n s l a t i o n a l cross sections, with d e f i n i t e momentum magnitude, have been described, the general formalism i s applicable to the scat t e r i n g of a r b i -t r a r y c l a s s i c a l and/or quantum states and to pure or mixed quantum systems including i n t e r n a l states. The l a t t e r only involves extra labels f o r the quantum states at t h i s formal l e v e l . A Gaussian momentum d i s t r i b u t i o n with dispersion K = kT^A (rather than the l i m i t 0) gives k i n e t i c cross sections. The present formalism gives some-hope of evaluating these d i r e c t l y , rather than averaging over i n d i v i d u a l , f i x e d momentum, c o l l i s i o n s - compare the discussion i n reference 13 > Connections to more standard formulations of c o l l i s i o n o theory have previously been discussed 7. CHAPTER 4 CLASSICAL TRAJECTORY APPROXIMATION 1 0 3 . 4.1.INTRODUCTION The c o l l i s i o n process has been described using a formalism that i s v a l i d f o r both c l a s s i c a l and quantum mechanics. This resulted i n an expression f o r the generalized cross section, Eq. ( 3 . 4 . 1 5 ) . which i s formally v a l i d i n both mechanics. In t h i s expression, the c o l l i s i o n i s assumed to have a well defined i n i t i a l momentum. The derivation of Eq. ( 3 . 4 . 1 5 ) began with an a r b i t r a r y d i s p e r s i o n ^ 2 i n the i n i t i a l momentum, and then required i t . to approach zero. The present chapter i s concerned with developing a semiclassical-type t r a j e c t o r y approximation that can be compared with the c l a s s i c a l cross section. For semicl a s s i c a l -type approximations, the l i m i t h-*0 of the quantal r e s u l t i s involved. As commented upon i n the fourth section of the second chapter, the order i n which the l i m i t s are taken i s very important. S p e c i f i c a l l y , the l i m i t h-*0 i s to be taken f i r s t , before the l i m i t "^p~*0. For t h i s purpose, i n the second section of t h i s chapter, a generalized cross section with non zero momentum dispersion i s defined by removing the 7\ ^T* ® l i m i t i n Eq 9 ( 3 . 3 . 3 ) . The r e s u l t i n g expression XT i s written as a time c o r r e l a t i o n function involving the superoperator 2^. The time dependence of t h i s c o r r e l a t i o n function i s generated by The exact c l a s s i c a l and quantal cross sections are recovered when the appropriate c l a s s i c a l and quantal quantities are used, see f o r example reference 1. This c o r r e l a t i o n function i s used as a basis f o r making semiclassical-type approximations. In t h i s formalism, 104. the differences between c l a s s i c a l and quantum mechanics occur i n two places, namely i n the superoperator ^ a n d i n the dynamical generator The differences between ">^ L and are interpreted here as s t a t i c quantum effects while the differences between expCi^^s) and exp(io^s) are dynamic quantum e f f e c t s . Two d i f f e r e n t s e m i c l a s s i c a l -type approximations are then envisaged, the f i r s t uses and O£Q while the second uses 2^ and C^Q^ The former emphasizes the dynamical quantum ef f e c t s but ignores any s t a t i c interferences a r i s i n g i n the c o r r e l a t i o n function. For t h i s reason, the f i r s t approximation w i l l be c a l l e d the dynamic interference approximation (DIA). Since exact solutions of the quantum dynamical motion are i n general unknown, the DIA w i l l not be pursued i n d e t a i l . The second approximation emphasizes the s t a t i c interferences a r i s i n g from the quantum nature of 2^ but takes the dynamics as the exact c l a s s i c a l phase space point to phase space point motion. This approximation w i l l be designated i n two d i f f e r e n t ways. To emphasize the s t a t i c nature of the quantum effects i n t h i s approximation ( i n contrast to the DIA), i t w i l l be c a l l e d the s t a t i c interference approximation (SIA). I t w i l l also be referred to as the c l a s s i c a l trajectory approximation (CTA) to emphasize the nature of the approx-imation to the dynamical motion. While equivalent, the usage of SIA or CTA w i l l depend upon whether the interference e f f e c t s or the dynamical effects are being emphasized. 105. The c o r r e l a t i o n functions f o r the exact c l a s s i c a l r e s u l t and f o r the CTA (SIA) are written i n terms of the exact c l a s s i c a l t r a j e c t o r i e s i n sections three and four respectively? The r e s u l t i n g expressions are compared and contrasted. The CTA cross section i s written as the sum of a formally positive term and a formally negative term.' This s p l i t t i n g of the cross section arises naturally from the nature of the superoperator ' 2 ^ . I t does not i n general constitute a s p l i t into a gain and a loss term as, f o r 2 example, done by Coombe et a l . The h-» 0 l i m i t of the positive term i s dominated by an expression proportional to *n~2 which can be considered as an extension to f u l l c l a s s i c a l t r a j e c t o r i e s of the semiclassical Mott • formula." In contrast to the Mott formula which i s a semiclassical t o t a l cross section, t h i s expression i s not a t o t a l cross section. The negative term i s dominated by the same express-ion i n the h -*0 limit,but, of course, of opposite sign. The c l a s s i c a l l i m i t of the combination of these two terms, the CTA generalized cross section, i s then the exact c l a s s i c a l r e s u l t s In the f i f t h section, the exact c l a s s i c a l t r a j e c t o r -ies are approximated by straight l i n e t r a j e c t o r i e s i n both the CTA and the c l a s s i c a l generalized cross sections. While the exact form of the s t a t i c interference e f f e c t s i n the SIA (CTA) depends upon t h e . e x p l i c i t ( c l a s s i c a l ) dynamics used, the existence of t h i s interference phenomena i s not affected by further approximations to the c l a s s i c a l dynamics.' 106. The straight l i n e approximation to the generalized CTA cross section i s shown to he equivalent to the usual Born cross section y Within the straight l i n e approximation, the positive term corresponds to the gain or t r a n s i t i o n Born cross section while the negative term i s the loss or t o t a l Born cross section. The equivalent straight l i n e t r a j e c t o r y approximation to the c l a s s i c a l cross section defines the c l a s s i c a l Born approximation. The c l a s s i c a l Born approximation i s a poor approximation since the p o t e n t i a l does not e f f e c t the t r a j e c t o r y . The differences between the SIA (CTA) and the c l a s s i c a l cross section a r i s e because of the differences between 2^ and 2^^. In concluding t h i s introduction, these differences are e x p l i c i t l y expressed by comparing the representations of and i n terms of the eigenfunctions of In the c l a s s i c a l case, using Eqs. ( 2 . 3 . 1 5 ) , ( 2 . 4 . 2 2 ) and ( 2 . 4 . 2 3 ) , the eigenfunction representation of becomes i n terms of the Fourier transform of the potential energy 107. (2|V|*') = ( 2 i r ) " 3 (drVCrJexp^ir.Cy-^^. ( 4 . 1 . 2 ) This rather cumbersome notation for the Fourier transform i s used because the Fourier transforms of "translated" potentials. ( ^ I v U + R ) ! * * ' ) = U n r ^ d r V U + R j e x p r-ix.CAr - X ' )~1, ( 4 . 1 . 3 ) for arbritrary R , are required in subsequent sections. In the quantal case, using Eqs. ( 2 . 2 . 2 3 ) , ( 2 . 4 . 2 2 ) and ( 2 . 4 . 2 3 ) , the eigenfunction representation of 2 Q^ becomes in terms of the same Fourier transform, Eq. ( 4 . 1 . 2 ) , of the potential energy. For the quantal case only, the round kets and bras of this notation are related to the Dirac kets and bras, |£) = n + 3 / 2|-n (4.1.-5) Classically, the round kets and bras have no significance whatsoever, and are just used as a convenient notational 108. device. In both the c l a s s i c a l and quantal cases, the eigen-function representation of 2 / i s non l o c a l i n the wave number, % , and i s proportional to the Fourier transform At of the poten t i a l i n the wave number difference. The difference between the two cases arises i n the v e l o c i t y dependence. C l a s s i c a l l y , the eigenfunction representation of l o c a l i n the v e l o c i t y , while quantally, i t i s non-local. In the quantal case, t h i s non-locality i n the v e l o c i t y i s coupled to the non-locality i n the wave number. This coupling leads to the quantal interference e f f e c t s between the two terms i n Eq. (4.1.4). These interferences are responsible f o r the s t a t i c interferences i n the SIA. F i n a l l y , the l i m i t h-»0 of the quantal expression, Eq. (4.1.4), leads d i r e c t l y to the c l a s s i c a l r e s u l t , Eq. (4.1.1), as required by the correspondence p r i n c i p l e . 109. 4.2.CORRELATION FUNCTION FORMULATION The generalized cross section with non zero momentum dispersion i s defined by removing the 0 limit i n Eq. (3.-3.3). ° ^ n < S > S ) = K 2 ) £ ^ g s c • ( * - 2 . l ) N i n c d R The usual generalized cross section, Eq. ( 3 e 4 . 1 5 ) , i s obtained by taking the limit 0 of Eq. ( 4 . 2 . 1 ) . For IT definiteness of presentation, the derivation in chapter 3 assumed the i n i t i a l free s t a t i s t i c a l state function was Gaussian. This condition i s now relaxed and the i n i t i a l free s t a t i s t i c a l state function i s l e f t arbitrary. As in Eq. ( 3 * 4 . 4 ) , the integrals over the time and the impact parameter are required (see chapter 3 for the notation), oo " , ( 2 ) dt < '06 d^'b f(r,p|t) =jdt J d ( 2 ) b f(r-pt^,p|0) — Oo =Jdt j d ( 2 ) b f[^(ptA)(g4 " P H ) - P 4"(p.p-t^),p|ol OO =y*/p.p" )jdz/d ( 2 )b f[r-z(p.$")pMy-p"P" )-zP" fp |0 J - t o o o = y*/p.p")Jdzjd ( 2 ) B f(r-zp",p|0) -oo •* = ^/P-P")gp.?(p). ( 4 . 2 . 2 ) As the above calculation shows, the time and impact param-eter integrations of the i n i t i a l free s t a t i s t i c a l state function produce a quantity proportional to the momentum distribution g*T(p) = fdr f(r,p | 0 ) (4 .2 .3) A/ having a mean p" and a standard deviation ^ . Following AJ P a procedure analogous to the development of Eq. (3»4.15), the generalized cross section with non zero momentum dispersion then becomes dv' * gen (g"-> R ) B - i ( 2 ^ ) 3 Jdw2 d, o This involves the transition superoperator, represented as a matrix in the eigenfunctions of The Miller superoperatorX Iwritten as a time integral, Eq. (3«4.?) , leads to an integral form for the transition superoperator o V* -ijds ^exp(ic^s) #exp(-i/^s). (4 .2.5) The f i r s t term of Eq. (4 .2.5) gives no contribution to the cross section, Eq. ( 4 . 2 . 4 ) . This i s due to the wave number dependences of Eq. (4.1.1) and (4.1.3). The cross section i s then given by the second term i n Eq. (4.2.5) CA O Cr^gen(P"-> R) = -(2Ty*)3Jdw2/d v'Ids ^ ( v R , 0 ) | ^ ^ e x p ( i / s ) ^ | ( v ^ O ) ^ ( v ^ > ) - 1 g ^ v • ) . * (4.2.6) which i s in the form of a time correlation function involving the superoperator 2^. The dynamical time dependence i s generated by oC . This expression i s valid for both classical and quantum mechanics. In this formulation of co l l i s i o n processes, there are two places where quantal vs. classical differences arise, 1 namely the differences between the superoperators and "2/^ , and the difference ,between the dynamical motions generated by ° £ a n d O ^ Q . The f i r s t has the appearance of a static quantum effect while the second modifies the time dependence, so that i t i s a dynamic quantum effect. Two different semiclassical-type approximations to the quantal case are now considered.* One obvious approximation i s to ignore the static interferences while retaining the dynamic; interferences. Thus, this approximation is termed the dynamic interference approximation (DIA). The DIA cross section i s thus defined OO o 112. X(vR,0)|^ C Lexp(i<s) ^ ( a | ( v \ 0 ) ^ ( X . . $ - ) 4 P ( / * v . ) . (4.2.7) This approximation differs from the classical case only because of the difference in the dynamical motions generated by C^TQ and ° £ Q - ^ * Further exploration of this approximation requires the evaluation of exp(io^s) and i s not pursued here. The other approximation involves replacing the complex dynamics generated by ot^ with the much simpler phase space point to phase space point classical dynamical motion generated by CL;; T h e s^ a-^° interference effects in are retained in this approximation. Thus, this approximation i s referred to as the static interference approximation (SIA). It i s also referred to as the classical trajectory approximation (CTA) because the dynamics has been taken as classical. Which of the two different names w i l l be used, w i l l depend upon which aspect i s being emphasized. The CTA cross section i s CO O ds 1 1 3 . The CTA can be evaluated with the f u l l classical motion, while further approximations to the classical dynamics can be made. Whether or not the f u l l classical motion i s used, the same types of. static interferences are present because these effects arise i n the superoperator 1^ and not i n the dynamics. The exact classical expression, Eq. (4.2.6), and the CTA, Eq. (4.2.8), are now discussed i n terms of the classical trajectories. 114. 4.3.EXACT CLASSICAL CROSS SECTION The phase space representation of the exponential of the classical generator, o ^ C I i , was given in chapter 2 "by Eqs. (2 . 3»H) and (2.3*12). The exact classical cross section involves the matrix element JV$r.»f * x p ( i / 0 I s ) ^ 0 L|(v',0)|> f j p ^ v - ) = ^/2n) 3JdVpr/Jdp'pv' exp(-ij<r) = 2 r r ) ~ 3 Jar jd*"' exp( - i r 7C)) <5"'l vfx+HsJ-rl/o)/- v~ fS/^v-Cs)! . (4.3.1) * P , P " defined by This expression contains the spread function f^s^tAv) fp" ( < A ^ ) = ^'^"rl4"i/i^)t (^.3.2) and the Fourier transform of the translated potential V^x+r(s)-rJ, see Eq. (4.1 .3). The position r(s) and the momentumyMy"(s) satisfy Hamilton's equations, Eq. ( 2 . 3 . 9 ) , subject to the i n i t i a l conditions r(0) = r and A I V " (0) = A V " 115. Using Eq.(4.3.1) i n Eq. (4 .2 .6) leads to the following expression for the exact classical cross section °* gen G L (£"-* R ) = J dPP 2 J d s / d ^ p ^ f £ <SM*> o ~ CO J J J exp(-ir ' (J-X"'))^ /%: _eL v[x+r(s)-r)| 0) -A- f * P ( p ( s ) ) | , ( 4 . 3 . 3 ) dp(s) p " Ax where the momentum pR i s defined "by p. The trajectories r(s) and p(s) satisfy the integrated forms of Hamilton's equations r(s) = r + jds' p(s')A* , ( 4 . 3 . 4 ) and ^ p(s) = p - (ds' iv(r(s*)) . ( 4 . 3 . 5 ) These classical positions and momenta satisfy time reversal symmetry, r(-s\r,p) = r(s|r,-p) ( 4 . 3 . 6 ) p(-s/r,p) = -p(s)r,-p), ( 4 . 3 . 7 ) as well as the parity conditions r(s r,p) = -r(s)-r.-p) AJ A* At Aj I At ^ ( 4 . 3 . 8 ) 116. and p(s r,p) = -p(s -r,-p). (4.<j3.9) ~ , LA# « J Al ^ Combining these conditions leads to the symmetry relations r(s|r,p) = -r(-sl-r,p), (4.3.10) and p(slr,p) = p(-s/-r,p). (4.3.11) Using these symmetry relations, as well as inverting the wave numbers £ and through the origin, leads to the symmetry principle (ds d £|dK*'/drexpC-ir'(X - X * ) ) (o| V fx) tf"7 flf « -p j V(x+r(s)-r)|o) T f ,?(p(s))\ J p I ~ 1 ~ " ~ ~ <)p(s) P - J AV *NI Tds|dx(d7<!'farexp(-ir .(K -K*) ) (o lv) f f ) : A f ( K J , / | v(x+r(s)-r ) | o ) ~~ f*?(p(s))\ . (4.3.12) J p / ~ ~ ~ ~ " <^ p(s) g - J A V ^ A» To obtain Eq. (4.3*12) the potential has been assumed to be spherically symmetric. The exact classical cross section i s thus written in terms of a time integral from minus to plus i n f i n i t y o - o o - J 0 mP w dK d r e x p ( - i r . ( # - X )) Ai A* 1 1 7 . ( 0 > [ * ) : 1 f(/lV(x+r(s)-r)|o) J - Ap(s))| * " J p / " ~ ~ ~ c)p(s) 5 - J L ( 4 . 3 e l 3 ) 118. 4.4. EXPLICIT CTA GROSS SECTION The CTA cross section was given formally by Eq. (4.2.8). It i s now given e x p l i c i t l y i n terms of the appro-priate classical trajectories. This cross section involves the superoperator product 2^exp(i«?^,Ls) In particular, the eigenfunction representation of 2 ^ Q , Eq. (4.1.4), has two terms of opposite sign. Thus, the cross section i t s e l f can then be written as a sum of two terms, one formally positive, the other negative. While this s p l i t t i n g has the appearance of being a division into the sum of a gain and a loss term, see for example Coombe et a l , this association does not hold in general. Howeverj within the Straight Line Approximation of Sec. 4.5» this association does hold. To obtain explicit expressions for these terms, i t is noted that Eq. (4.2.8) involves the matrix element = (2ir J'^rJa^jdv-expC-ir'TC+iris)*") - •fe-1(2w33 fdrTa*"'' exp(-ir. ("K- K (*'"/ V( x+r( s )-r) / 0) f t v" (s 1$' ) /// )) /// )119. #L . /// 7 " f p ^ / ^ s ^ f f >J • ( 4 . 4.1) The formally positive term i n the CTA cross section i s then e x p ( - i ( * - * )-r)(0 V X) / ( * /V(R +(s)+x-r) 0) fJ?(P (s)+tfX ) + (5 / V(x+R-(s)-r)/0)f p ( P - ( s ) - i t e ) , (4.4*2) while the formally negative term i s e x p ( - i ( * - K % r ) ( o | v | * ) f«fiv(x+R+(s)-r)J 0) f ! P ( P + ( s ) - r f / ) + (/1v(x+R-(s)-r) /0)f5?(r(s)^/)j •. P .A, /v *v A , I <V p v _J ( 4 . 4 . 3 ) The CTA cross section i s the sum of these two terms. The positions R +(s),R7(s) and the momenta P,+(s) ,P~(s) satisfy Hamilton's equations respectively with the i n i t i a l conditions R + (0 ) = r, P, +(0) = pR + £iX and R~(0) = r, P"(0) = pR JS The second part of the positive term satisfies a symmetry principle similar to the classical term. Inverting the wave numbers K and ^ "'through the origin leads to o fds d^|d/ //drexp(-i(^- * " ) . r ) ( o / v / * ) J A A^ J Ay A*" Ay * » ' 120. ^Jdsjdjjdft'' Jdrexp(i( £- ?"Vr)(o|v|*) (X /'7v(x-R+(s)+r)//0) f p P ( P + ( s ) H n / ' ) , (4.4.4) where the s p h e r i c a l symmetry of the p o t e n t i a l has a g a i n been used. The c l a s s i c a l p o s i t i o n R + ( s ) and the momentum P + ( s ) s a t i s f y combined symmetry r e l a t i o n s s i m i l a r t o r ( s ) and p ( s ) , Eq. (4.3.10) and (4.3.11). The symmetry p r i n c i p l e f o r the second p a r t of the p o s i t i v e term becomes JdsJd* | d f c " ' J d r e x p ( - i ( £ - )• r ) (0\V[K) •oo . (K'"I V(x+R-(s)-r)/o)f'' p(p-(s)-ife7/ /) Ay =jfds111< jdf< Ydrexp( - i (X - • r) (0) v|*) ($'7 V(x+R+(s)-r)/ 0 ) / p ( P + ( s ) - i n X / / / ) . (4.4.5) The positive part of the CTA cross section i s then given in terms of an integral over the time s from minus to plus i n f i n i t y oc 0 0 121. exp(-i(^-^)-r)(0)v|K)(W>(x+R+(s)-r)|0) fJ?(P+(s)+*fc2'''>. ( 4 . 4 . 6 ) The second term of the negative part of the CTA cross section, Eq. ( 4 . 4 . 3 )« also satisfies a symmetry principle similar to Eq. ( 4 . 4 . 5 ) so that the negative term i s given by an expression similar to Eq. ( 4 . 4 . 6 ) , namely •oo exp(-i(2C-^).r)(0)v|5<)(K"7 V(x+R +(s)-r)|0) fJ?(P ( s ) - r f i £ ) • ( 4 . 4 . ? ) The generalized CTA cross section i s the sum of Eqs. ( 4 . 4 . 6 ) and ( 4 . 4 . 7 ) , exp(-i ( J - y )^r)(o/v/* )(*''/ V(x+R +(s)-r)/o) f J?(P+(s)+tf - f J(P +(s)-4 ^ A W ) >. ( 4 . 4.8) P - V p -Ay A . I A / <•»/ The difference between the CTA generalized cross 122. section and the exact classical cross section, Eq. (4.3.13)> i s due to the static interference effects inherent in but which are lacking in The non-locality i n the velocity dependence of &*Q, which is coupled to the wave number dependence, see Eq. ( 4 . 1 . 4 ) , results in the i n i t i a l momentum of the classical trajectory being P (0) = pR + inK rather than simply p(0) = pR, which i s the case for the exact classical cross section. Thus, the effect of the static interferences i s to include an average over different classical trajectories. The classical limit of the CTA cross section i s now discussed. In particular, the limit h-»0 of the positive and negative terms are considered seperately. Both these terms are dominated by an expression proportional t o * ^ - 2 ; but, of course, of opposite sign. In the next section, this expression i s shown to reduce to the Mott -'formula when the classical trajectory i s taken as a straight line. The f u l l CTA cross section reduces to the classical expression, Eq. ( 4 . 3 . 1 3 ) , i n the limit h-> 0. The CTA cross section involves the classical trajec-tories R +(s) and P,+(s) which depend upon Plancks constant through the i n i t i a l condition P + ( 0 ) = pR + i n * = p + i"n£. For small quantum effects these might be approximated as R +(s) ~ r(s) -Hr£*-4- r(s) + i^/z)2^'- iJL r( h - 0 ~ o>p ~ <?p)p~ = r(s) + i l k + ( * i 2 / 8)b, , ( 4 . 4 . 9 ) 123. and P +(s) ~ p(s) +fn*-A p(s)+i(*n)27<2r : ^ _ p(s) h-*0 ~ 3p ~ ip^p ~ = p(s) + i*ha„ + (n 2/8)b , (4;«4.10) •to «•>/ to second order i n n, with the vectors a .a ."b and b.. - r ~p -r -p * * * * * * «v* ' being defined appropriately. Thus only the classical trajectories r(s), Eq. (4.3.4), and p(s), Eq. (4.3.5) are required. The trajectory R +(s) only appears i n the Fourier transform of the translated potential V(x+R +(s)-r). As h-*0, this Fourier transform becomes iff I V(x+R +(s)-r)| 0) h-» 0 (K"/ V(x+r(s).£+i*i9r+*2br/8)^  0) ^ 0 (?">(x+r(s)-r)/0) Jl+iitxaJ.5/,/: + (n 2/8) [ i V K Y " -a^./fV*] } , (4.4.11) *» A. A / A / J to second order in*n. To obtain Eq. (4.4.11), the potential was expanded in a Taylor series about the position x+r(s)-r : and then an integration by parts performed. The CTA cross section terms also involve the spread functions f* p(P +(s)+fnfc") and f*?(P+(s)-£n£*) which, when P «V Ay P V A< A» expanded i n powers of n become 124. , , /// K_ in , „ (s)±iH< ) — f P(p(s)+in(a + *)+n 2b / 8 ) I? h-»0 £ ~ £ ~5 ^ f!?(p(s)) + in*(a +/)' — / P ( p ( s ) ) h-»0 £ - ~P * ^p(s) 5 -+ ^ 2 / 8 ) [ b ? - i ( s ) fJ ( £ ( e H«pts')( S ?t5'). , f^ p(p(s))? , (4.4.12) M s ) >p(s) P - J to second order i n "n. The dominant contribution to the limit h-*0 of the positive term is then -op exp(-i (y-^' / /)-r)(0|v|}:)(^/v(x+r(s)-r)/0)fpP(p^ T 4.4.13) When a straight line trajectory approximation is made to this term, Sec. 4 . 5 , the Mott ^ semiclassical approximation to the total cross section results. Eq. ( 4 . 4 . 1 3 ) can be considered as an extention of the Mott approximation to involve the complete classical trajectory. In contrast to the Mott formula, this i s not a total cross section. The dominant contribution to the negative part of the CTA cross section i s the negative of Eq. ( 4 . 4 . 1 3 ) . The f u l l CTA cross section involves a difference of the spread functions which, to second order i n i i , i s * a, A" , . f t fj?(p (s)+ihX ) f P(p + ( s ) - iK£) 125. ^ ^ ^ L o r- /-/ \^  f »(p(s),) h-0 * ^p(s) P ' L"-£ ~r J ^p(s)^p(s) ~ (4.4.14) The f u l l CTA cross section in the limit h-»0 i s then, using Eqs. (4.4.11) and (4.4.14), g e n - h-*0 J exp(-i(* -$%r ) ( o l v | X ) ( * f f ' / V(x+r(s)-r)/o) Jp(s) J ~ Jp Jp(s) Jr ~ + X * • * - p ( s ) : i £ f p ( p ( s ) ) [ . ( 4 . 4 . 1 5 ) * ~ }p~ ,)p(s)Jp(s) P - J The term proportional to n" 1 i s identically zero due to the symmetry principles, see for example Eq. ( 4 . 4 . 5 ) . The remaining terms are seen to be identical to the terms in the classical cross section, Eq. ( 4 . 3 . 1 3 ) , after the gradient with respect to the p in the later formula i s A V reexpressed,namely $ " % ! T f(x"'V(x +r( S)-r)(0) i - A j l . ) ) ] dV L J P(s) Av J A . = (/Vv(x+r(s)-r)/0) jX /X: i _ p ( s ) . i ! f* P(p(s)) * V '~ JP - ^ P < S ) ; P ( S ) P -+ i x " ^ : i. r ( s ) - x " i _ f P(p(s))( . ( 4 . 4 . 1 6 ) " ^P < M s ) P - J A . Rearranging the dot products in Eq. ( 4 . 4 . 1 6 ) leads to the 126. non vanishing terms in the curly brackets in Eq. (4.4.15). Thus the classical limit of the generalized CTA cross section i s the exact generalized classical cross section. 1 2 7 . 4.5. STRAIGHT LINE TRAJECTORY APPROXIMATION AND THE  BORN CROSS SECTION The present formalism has lead to the introduction of two approximations, the DIA and the SIA (CTA). The SIA involves the quantum superoperator while the dynamical motion i s generated by the f u l l classical Liouville super-operator C L . The differences between the SIA and the exact classical cross section are due to the static inter-ference effects contained i n 2^. While the exact form of the static effects in the SIA w i l l depend upon whether the explicit classical dynamics used i s exact or approximated, the existence of this interference phenomena i s not affected by the type of dynamics used. For simplicity, the trajectories are now approximated as being straight lines. This straight line motion can be interpreted as a particular high energy approximation to the true classical dynamics. This approx-imation i s called the straight line classical trajectory approximation (SLCTA). The resulting SLCTA cross section is shown to be equal to the usual Born cross section. A similar straight line approximation to the exact classical cross section (or equivalently to the DIA) defines the Classical Born approximation. This straight line approx-imation i s also applied to the dominant term in the classical limit of the formally positive term in the CTA cross section. The resulting expression i s equal to the Mott J semiclassical formula. 128. 4.5.a. THE SLGTA The CTA cross section involves the classical position R +(s) = r +/*."1^ds-« P +(s'), (4.5.1) and the classical momentum P +(s) = pR + -^ds-' >V(R +( S')) . ( 4 . 5 . 2 ) The simplest approximation to these trajectories i s to neglect the derivative of the potential, that i s , the force, in the momentum equation. This leads to the straight line trajectories P + ( s ) S L = pR + in£ , ( 4 . 5 . 3 ) and R + ( s ) S L = £ + (pR+rf$)s/^. . ( 4 . 5 . 4 ) These constitute a straight line approximation to the exact classical dynamics. This approximation may he expected to be valid i f the magnitude of the time integral over the force i s much smaller than the magnitude of the i n i t i a l momentum. This should be the case for large i n t i t i a l momenta which implies that this approximation can be considered as a high energy approximation to the classical dynamics. The position trajectories R (s) and R (s) are depicted i n Fig. 4.1. Using Eqs. ( 4 . 5 . 3 ) and ( 4 . 5 . 4 ) i n the expression 1 2 9 1 3 0 . for the positive part of the CTA cross section, Eq. ( 4 . 4 - . 6 ) , leads to the positive part of the SLCTA cross section (with non zero momentum dispersion) OO <* v~»> • H* H dr - c o exp(-i(£-5 )-r)(0|v|^)(x"j V(x+(pR+in £ )s/^ )\o) f p.. . x - ^ OO = (2TT ) 3h" 2jdpp 2jdsjd^ (0|v|x)(«/v(x+(pRH^)s^u)|0) f^?(pR+n*). ( 4 . 5 . 5 ) The negative part of the SLCTA cross section i s 0 6 0 0 ^ o - o o J (?l V(x+(pR+in J ) s / > ) i 0 ) f ^ p R ) , ( 4 . 5 . 6 ) while the generalized SLCTA cross section i s the sum of Eqs. ( 4 . 5 . 5 ) and ( 4 . 5 . 6 ) . The positive part of the SLCTA can he written i n terms of Dirac kets and bras, 00 <*> crJ-fJfV^R) - (2Tr)3t;-2|dpp2|dsJdy(e/v/j) 131. ft (*IVIp/)exp(iK- (pR+tf * )s/>) )f p p(pR+n$ ) /* = (2Tr)3n-5Jdpp2Jdp' |( PR/h |v lp'/n)) 2 / P(p') jdsexp(i(p ' -pR) ' (p' +pR)s/2nyU,) = ( z i r ) > 2 f i 2 [ d p - p ' ( p ' . > ) - ^ < p ' R | v | p - > | 2 g p , ? ( p ' ) = Jdp' ( p , , - p , r 1 ^ t ? a n a ( p ' - * R ) S p P ( P ' ) - (^-5.7) This i s recognized as an average over the usual expression for the Born transition cross section . In a similar manner, the negative part of the SLCTA cross section, Eq. ( 4 . 5 . 6 ) , i s an average over the usual Born approximation to the total cross section. Thus, i n the SLCTA, the sp l i t t i n g of the generalized cross section into a positive and a negative part coincides with the s p l i t t i n g into a gain and a loss term. The straight line trajectory approximation to the CTA results i n an expression for the cross section, the SLCTA cross section, which i s equal to the Born cross section. In this manner, the CTA can be considered as an extension of the Born approximation i n which the straight line trajectory i s replaced by the f u l l classical trajectory. It is an extension to the generalized Born cross section as opposed to an extension of the transition or total Born cross sections. 132. 4.5.b. CLASSICAL BORN APPROXIMATION For comparison, a straight line approximation i s made to the exact classical cross section which i s thus by-nature a classical Born approximation. The exact classical cross section involves the position r(s) and the momentum p(s) given by Eqs. ( 4 . 3 . 4 ) and ( 4 . 3 . 5 ) . As with the trajec- _ ~ tories R +(s) and P + 0 s ) , the straight line approximations are p ( s ) S L = pR, ( 4 . 5 . 8 ) and r ( s ) S L = r + pRsAc . ( 4 . 5 . 9 ) A / AX' / SL The position trajectories r(s) and r(s) are depicted i n Fig. 4 . 2 . The only difference between r(s) and R (s) , and between pts) 3*' and P +(s)^ I i, is the presence of the additive quantity K i n the SLCTA momentum. This arises from the static intereference effects of It i s a crucial difference. Since there i s an integral over the wave number *K , then a l l observable angles are sampled, in the SLCTA transition cross section, due to the wave number dependence of the spread function. But, as i s to be shown, in the classical case the spread function does not contain a wave number dependence. The result i s that only those observation angles centered about the forward direction contribute to the classical Born approximation. Using Eqs. ( 4 . 5 . 8 ) and ( 4 . 5 . 9 ) in the exact classical cross section, written i n terms of the trajectories, Eq. ( 4 . 3 . 1 3 ) , the 1 3 4 . classical Born or straight line trajectory cross section be c orae s oo a o Or g^ n C L B(P M-»R) * *Jdpp2JdsJdff(d*f (drexp(-i(X-*").r) (P R) = 4Tr 3Jdpp 2JdsfdX (o|v|*)X7< S ^ . A fcxlvCx+pRsAtjIo) A A p f i ) ? . ( 4 . 5 . 1 0 ) The f i r s t gradient with respect to pR operates on the product of a matrix element involving pR and the gradient of the spread function. Operating on the matrix element, the gradient gives ^ ( t S \ v ( x + P R s ^ ) \ 0 ) = (2TT )"~3fdxV(x) v — ^ exp(-i(x-pRs/u)-£ ) J ~ ~ <)(pR> " / = isyuT1 ( * | V(x+pRs^)) 0 ) . ( 4 . 5 . 1 1 ) The classical Born approximation i s then the sum of two terms ** ? ° / g e A G L B ( £ " ^ R ) 88 ^3/dpp2Jdsfd/<(0|v))k) o -co CK 1V(x+pRs^) I 0) £ is^TllC 2 * j ~ ^f p>( pR) 1 3 5 . On defining the position z = ps/^c, ( 4 . 5 . 1 3 ) the classical Born approximation becomes ° " g e A C L B (P"^ R ) = ^ j f d s / d g (2lWt5)%| V(x+zR))0) (4.5.14) The integrals over z can be calculated to give 00 Jdz z (* ) V(x+zR))0) — 0 0 = - i (2IT) V(K) <)£(#,,) ,< ( 4 . 5 . 1 5 ) and ^ Jdz (g/v(x+zR)/o) = (21T) % ) £ ( * „ ) , ( 4 . 5 . 1 6 ) - 00 where the wave number has been written in terms of the component Knparallel to R and > x^, perpendicular to R. The Fourier transform has been written in the more conventional notation (*|v|o) = V(X) = (2TT )~ 3 drV(r)exp(-ir-X ). ( 4 . 5 . 1 7 ) Ay Ay J y v y W y w / V 136. With these evaluations, the classical Born approximation i s then OO CH = S T T^uJd** 3 | V ( X ) | 2 ( U - r a ) i - * | g j d p p - 1 f ^ ( p 8 ) . (4.5.18) The angular dependence of the classical Born approximation is given by the double gradient of the integrated spread function J d p p - 1 f p P ( p R ) =/*-(p"-R)" 1Jdpp" 2 gp P(pR). (4.5.19) For this approximation to be f i n i t e upon integration over - 2 XV)/ ' V the momentum, p g s(pR) must be an integrable function of /V p. In chapter 3/ a Gaussian momentum distribution function was used as an example. Here i t i s an inappropriate choice since p times a Gaussian i n momentum diverges at the origin. One way of remedying this condition on the momentum distribution i s by introducing a low momentum cutoff. Since the incoming momentum distribution i s centered about the incoming momentum direction p", then the classical Born approximation i s also centered about this direction 137. with the exact dependence given by the double gradient of the integrated spread function. The exact classical dynamical motion has been replaced by a straight line trajectory i n this approximation. Even though this straight line trajectory i s i n the direction of the observable A . angle R, see Fig. 4.2, the classical generalized Born cross section i s s t i l l centered about the incoming direction p". Thus, i t i s more like an approximation to a total cross section than to a transition cross section. The explicit form of the potential plays no role (other than introducing an overall multiplicative constant) in the cross section. It does not affect the angular dependence of the cross section nor does i t affect the energy dependence, which i s determined solely by the form of the incoming momentum distribution. Physically, the particle may be thought of as travelling so fast that the potential has no effect on the i n i t i a l free linear trajectory of the particle, i.e. no scattering. Any angular dependence of the cross section i s due to the angular dependence of the i n i t i a l momentum distribution. In the case where the momentum distribution becomes a Dirac delta function, Eq. (4.5.19) is / d p p ^ A p R ) - ^ ( R - P " ) . ft.5.20) 138. while the classical Born approximation to the generalized cross section i s Because the incoming momentum i s precisely known i n this case, the cross section i s in the forward direction only and formally i n f i n i t e . As in the more general case, Eq. (4-.5.18),1 the dependence of the cross section on the incoming momentum magnitude i s governed only by the incoming momentum distribution and i s independent of the explicit form of the potential. In this case the momentum dependence is the inverse fourth power which diverges as the incoming momentum magnitude p" approaches zero. The reason the classical Born approximation i s not a good approximation is because the trajectory i s too crudely approximated; that i s , the momentum of the trajectory i s in the observable direction R in contrast to the SLCTA (or Born) where the direction of the straight line trajec-C g f l p ' ^ R ) = 8 l T 5 ^ 2 ( p " ) - 4 / d * * 3 | v ( x ) | 2 (4.5.21) tory involves the wave number \K, 139. 4 .5.C. SEMICLASSICAL MOTT FORMULA Finally, the dominant term in the classical limit of the positive part of the cross section i s evaluated in the straight line approximation. Using Eqs. ( 4 . 5 . 8 ) and (4 .5»9) i n Eq. ( 4 . 4 . 1 3 ) and taking the momentum dis-persion to zero, gives-00 CO v r J dx V ( r ) e x p(i ( r - x W < ) "OO v(r)(«fv(x+pRs/^)/0)^a/p") S(pR-p") = UTrt^P'^n^Jds dX Idr - 0 0 ^ V(x+p"Rs/J«.) ?(R-p"). ( 4 . 5 . 2 2 ) Writing the vectorial quantities with components parallel A- A to R and perpendicular to R, this leads to OO dzV(zp"+b) 2 , ( 4 . 5 . 2 3 ) which i s i n the form of a total cross section times a Dirac delta function. This evaluation of the total cross section i s equal to the Mott J semiclassical formula for the total cross section. Thus, Eq. ( 4 . 4 . 1 3 ) can be considered as an extension of the Mott formula to involve the complete classical trajectories. In contrast to Eq. ( 4 . 5 . 2 3 ) i t i s 140. not a total cross section. 141. 4.6. DISCUSSION The generalized cross section with non zero momentum dispersion has been written i n terms of a time correlation function, Eq. (4.2.6), for both classical and quantum mechanics. Differences i n the classical and quantum cases arise in two places, namely in the static and i n the dynamic interference effects. Thus two obvious approximations to the quantal cross section presented themselves. The DIA emphasizes the quantal dynamic interferences while suppressing quantal static interferences. The SIA emphasizes the quantal static interferences while the dynamics are taken as classical. For this reason, the SIA was also called the classical trajectory approximation (CTA). The CTA was pursued i n some detail, as summarized i n Table 4.1. In particular, the CTA was written as a sum of two terms, one formally positive, the other negative. This s p l i t t i n g arose naturally from the explicit form of the quantum superoperator 6*^  and i s , i n general, different from the usual gain and loss formulation, see for example, Coombe et a l . Upon assuming a straight line trajectory, the CTA cross section became the corres-ponding generalized Born cross section. In this case, the positive term was equal to the gain or transition Born cross section while the negative term was the total Born cross section. The h-^0 limit of the positive part of the CTA generalized cross section led to a f u l l classical trajectory generalization of the Mott 3 semiclassical formula (GMOTT). The classical limit of the generalized •QM(Gen) DIA(Gen)-*- SIA(Gen)=GTA(Gen) • oo •< h = 0 GMOTT ( + ) t h < < 1 GTA( + ) SLA oo < h"° MOTT SLA (Tot)< h < , < 1 SLCTA (Tr) CTA(-) h < < j GMOTT(-) h = $ SLA j SLA SLCTA (Tot) — — — i - -MOTT (Tot) h=0 - oO - PO •h=0 SLCTA(Gen)=Born(Gen) h=0 h=0-CM(Born) SLA CM(Gen)-*-Table 4 . 1 . 143. Legend for Table 4.1. Gen = Generalized Gross Section Tr = Transition Cross Section Tot = Total Cross Section (+) = Positive Part of the CTA (-) = Negative Part of the CTA 144. SLCTA cross section defined a classical Born approximation. The CTA (and DIA) are semiclassical-type approximations, hut not true semiclassical approximations since parts of the cross section are selectively expanded in 1i while other parts are retained quantally. The relationship "between the f u l l CTA and standard semiclassical wave function approx-imations ^ is not at present known. An elucidation of any such relationship requires further work. The CTA was evaluated for the case of a straight line trajectory. Other trajectory approximations are possible. If the momentum i s not a constant, then the appropriate classical cross section w i l l not suffer the same fate as the classical Born approximation. It i s also hoped that the CTA w i l l provide an alternate approach to the standard methods ^ for studying the classical i n f i n i t i e s i n the classical cross sections rainbows, glories and orbitings, together with their appropriate semiclassical structures. CHAPTER 5 DOUBLE POTENTIAL SCATTERING: DISTORTED WAVE BORN AND SEMICLASSICAL-TYPE APPROXIMATIONS 146. 5.1.INTRODUCTION The formulation of chapter 4 resulted from the explicit form of the generator c*^. Por that development, the generator was taken to be the sum of a free part and a part due to the potential, 1f\ Using the *K motion as a reference, the cross section was expanded in powers of The dynamical motion can also be expanded about some other reference behaviour, in particular that generated by the sum of the kinetic hamiltonian K and a non-zero potential, say V Q (called the reference potential). Again, this development can be done for both classical and quantum mechanics. In this case, the generator i s again written as the sum of two terms, one term being the part which generates the reference motion due to K and V Q , ^ Q = ^ » and the other part being the difference, 2/^  = o£- <^Q» The quantal generalized cross section for such a double potential formulation has been investigated by Snider and Turner . This double potential formulation i s now re-investigated in light of the unified description of classical and quantum mechanics presented in chapter 2. 1 Snider and Turner wrote the quantum generalized cross section in terms of a parameterized transition super-operator (in contrast to the time integral form of the transition superoperator). To reduce the superoperator expressions to the standard operator formulas, an assumption about the nature of the parameterized transition superoperator 14?. was required. The present treatment avoids this assumption by dealing exclusively with the time integral forms of the transition (and Miller) superoperators. These integral expressions are given i n the second section and are compared with the parameterized transition superoperators* In the third section, the cross section i s written as a sum of three terms,1 the f i r s t being the reference cross section, the second being f i r s t order i n 7^ and the third being the remainder. In contrast to the Snider and Turner expressions, the cross section given here i s valid for both classical and quantum mechanics. The remain-der i s written i n the form of a time correlation function involving the superoperators ^•S^jJ and £Ll 2S. (-^-T. o o o being the transpose of the Miller superoperatorSX ^  )• o The dynamical motion of this correlation function i s generated by the f u l l interaction superoperator This time correlation function i s compared to the time correlation function of chapter 4 which i s the case for a single potential. Snider and Turner have shown that replacing the f u l l dynamical motion by the reference motion (generated by in the quantal time correlation function gives an expression for the cross section equivalent to the standard transition operator formulas for the distorted wave Born approximation (DWBA). It i s i n this reduction that the assumption concern-ing the parameterized transition superoperator was used. This 148. reduction i s repeated i n section four, but, now using the time integral formalism where no assumptions are required. In section five, the exact double potential classical cross section i s written i n terms of the complete classical trajectories. Replacing the complete trajectories by the reference trajectories defines the classical DWBA approx-imation to the generalized cross section. Since this approximation involves the classical reference scattering superoperator, i t does not suffer the same fate as the classical Born approximation. In the sixth section, semiclassical-type approx-imations to the exact quantal generalized cross section are presented. As in chapter 4, there are two types of differences which occur between quantal and classical mechanics; namely, static and dynamic interferences. This leads naturally to two different approximations. One approximation i s the dynamic interference approximation for double potentials (DDIA) while the other i s the static interference approx-imation (DSIA). As in chapter 4, the DDIA i s not pursued. A further approximation to the DSIA i s presented in section seven where a l l the effects of the reference potential are assumed classical. This approximation i s called the classical trajectory approximation for double potentials (DCTA). It i s presented in terms of the appropriate classical trajectories. The double potential classical cross section results when the f u l l DCTA cross section i s evaluated i n the limit h-*0, while the DWBA to the DCTA results when the complete 149. trajectories are replaced by the reference trajectories. Only translational motion i s presented in this chapter. In chapters 6 and 7, internal states are included. Kinetic cross sections are defined and evaluated there, using the double potential formulation of this chapter. 150. 5.2.M0LLER AND TRANSITION SUPEROPERATORS FOR DOUBLE  POTENTIALS 5.2.a.ABSTRACT AND PARAMETERIZED SUPEROPERATORS Linear transformations from one s t a t i s t i c a l state to another s t a t i s t i c a l state can be described through the use of superoperators. For example, the interacting s t a t i s t i c a l state of a scattering event, S(t), i s related to the i n i t i a l free s t a t i s t i c a l state, s(t),through the Miller superoperator, S(t) =IL Ls(t). ( 5 . 2 . 1 ) The Miller superoperator, Eq. ( 2 . 6 . 4 ) , can be expressed i n the form of a time integral,Eq. ( 3 * 4 . 7 ) , SI, = lim exp(i<^t )exp(-i^t) = 1 - i J d s e x p U ^ s ) Z A e X p ( - i ^ s ) . (5*2 .2) - oo Eq. (5*2.1) can be expressed in parameterized form i f the eigenfunctions of are used. In terms of the expansion coefficients f y ^ ( t ) of s(t),see Eq ? ( 2 . 4 . 2 6 ) , the s t a t i s t i c a l state S(t) i s s(t) = lim [dvC;<nL(x.X+i?)( (v.fjjk f y f c ( t ) . * " ( 5 * 2 . 3 ) where the parameterized Miller superoperator i s 151. ( 5 . 2 . 4 ) Eq. ( 5 . 2 . 3 ) i s obtained from Eq. ( 5 . 2 . 1 ) when the second form of Eq. ( 5 . 2 . 2 ) i s used and when the small parameter i s introduced to insure convergence of the time integral. 1 Snider and Turner used the parameterized form in their analysis of the double potential formulas. This analysis required an assumption about the ^  dependence of the parameterized transition superoperator. The need for this assumption can be avoided by using the time integral forms for the Miller and transition superoperators as i s done in this chapter. Explicit time integral forms for the Miller and transition superoperators are now presented for the case of a double potential. The equivalent param-eterized forms are also given. 152. 5.2.b.M0LLER SUPEROPERATOR The f u l l Miller superoperator,Eq. ( 5 . 2 . 2 ) , can be written i n terms of the reference Miller superoperator, 1 1 , = lim exp(ic< nt)exp(-i ), ( 5 . 2 . 5 ) O t-*-Oo the superoperator 2^ and the f u l l generator oC. To do so, use i s made of the identity s exp(i^s) = e x p ( i ^ s ) +i ^ d t e x p ( i o 2 ^ e x p ( i ^ ( s - t ) ) . ( 5 . 2 . 6 ) In particular, this identity i s used in the 2J^ part of Eq. ( 5 . 2 . 2 ) , 9 SI L = l - i y d s e x p ( i ^ s ) ( ^ ) e x p ( - i ^ s ) = _QL - i ydsexp(ic^s) Z^exp(i^s) o -Oo - i ^ d s ^ d t e x p ( i ^ t ) ^ e x p ( i ^ ( s - t ) yoexp(-iTs). ( 5 . 2 . 7 ) The order of integration i n the double integral i s changed, 153. /ds Jdt = - Jdt Jds, (5.2.8) and the time s i s translated to s-t. The r e s u l t i n g express-ion f o r the M i l l e r superoperator i s o -TL = XL -i/dsexp(i<^s) V^Q-^ exp(-i/^s). L L -co o 0 ( 5 . 2 . 9 ) The i n t e r a c t i n g s t a t i s t i c a l state S(t),Eq. ( 5 . 2 . 1 ) , i s then given i n terms of the reference s t a t i s t i c a l state S Q ( t ) and a time i n t e g r a l , o S(t) = S' 6(t) - i J d s e x p j i o f s ) Z^S o(s+t). ( 5 . 2 . 1 0 ) This equation can also be expressed i n parameterized form when the eigenf unction representation (of t^ C) i s used. In pa r t i c u l a r , a two parameter form a r i s e s , S ( t ) ^ n L ^ ' ^ e ) l ^ ^ f Y t ^ ^ i*o+ (5 .2 .11) where the two parameter M i l l e r superoperator i s ^ L ( r £ ' l ' £ ) = (i+(y-5-ote7+i£)"1c^1)fLL (v.x+i^). ( 5 . 2 . 1 2 ) The r i g h t hand side of Eq. (5 .2 . 11 ) i s equal to the rig h t hand side of Eq. ( 5 . 2 . 1 0 ) only i n the double limit^->> 0 + £ - * 0 + , on the assumption that the l i m i t s are well behaved. 154 On the other hand, a sin g l y parameterized M i l l e r superoperator, Sljiv-wj) = ( i ^ f - / t i 7 r ¥ 1 ) i i L ( v - * + i ? ) , ( 5 . 2 . 1 3 ) i s obtainable from Eq. ( 5 . 2 . 4 ) . The i n t e r a c t i n g s t a t i s t i c a l state,Eq. ( 5 . 2 . 3 ) , i s then S(t) = lim^jdyjd& ( l + ( y # - . A i ^ ) " 1 l^) - f 2 L 0 t * s t l r > | ( i ' S j ^ ' y > s ( t ) - (5.2.1*) The r i g h t hand side of Eq. (5.2.14) i s equal to the right hand side of Eq. ( 5 . 2 . 1 0 ) only i n the l i m i t ^ - * 0 + . I t i s also equal to the r i g h t hand side of Eq. ( 5 . 2 . 1 1 )jagain, only when a l l the l i m i t s are taken. This suggests that the parameterizations of the resolvent and the reference t r a n s i t i o n superoperator i n Eq. ( 5 . 2 . 1 3 ) are independent when s% approaches zero. This i s exactly the type of assump-t i o n needed by Snider and Turner to obtain the quantal DWBA operator formulas from the superoperator expressions. Eq. ( 5 . 2 . 9 ) f o r the time i n t e g r a l form of the M i l l e r superoperator i s now used as a s t a r t i n g point f o r obtaining a double potential time i n t e g r a l form f o r the t r a n s i t i o n superoperator. In Sec. 5«3,this time i n t e g r a l form of the t r a n s i t i o n superoperator w i l l be used to obtain an expression f o r the generalized cross section which i s 155. valid i n both classical and quantum mechanics. 156. 5.2.c.TRANSITION SUPEROPERATOR The abstract transition superoperator, can be written as the sum of two terms, 7= y 0 * x i i s t . (5.2.16) when use i s made of Eq. ( 5 . 2 . 9 ) . The f i r s t term i s simply the abstact transition superoperator for the reference system, y0 = 2*0QL . ( 5 . 2 . 1 7 ) 0 while the "distorted"(for lack of a better description) transition superoperator involves a time integral, o X i s t = ^ L o - i / d s ( ^ 1 + ^ 0 ) e x p ( i ^ , — 00 2^±9.L e x p ( - i ^ s ) . (5.2.18) The latter can be rewritten using the identity s exp(i^s) = expdS^s) +ijdtexp(i©fct) o Z^expt i t^ ( s-t)), (5.2.19) which,in contrast to Eq. ( 5 . 2 . 6 ) , has the f u l l superoperator to the right of the superoperator i n the time integral. When Eq. (5.2.19) is used i n the 2^ part of the time 157. i n t e g r a l i n Eq. (5 .2 .18), the distorted t r a n s i t i o n super-operator i t s e l f becomes the sum of two terms, ^ d i s t 88 ^1 +^R' ( 5 . 2 . 2 0 ) The leading t e r m , ^ ^ i s f i r s t order i n the superoperator 2^, -X = ^ S ^ L " i J d s ^ 0 e x p ( i 0 ^ ) s ) ^ ; L / 2 L e x p ( - i ^ s ) , O o ( 5 . 2 . 2 1 ) while the remaining term i s of second order and higher orders i n o J7 R = -i/ds ^ 1 e x p ( i / s ) ^ 1 i 2 L exp(-i/ts) o S - i 2 Jdsjdt y Qexp( i ^ t ) ^ e x p ( i / ( s - t ) ) ^-QL - O o 0 0 e x p ( - i ^ s ) . ( 5 . 2 . 2 2 ) The remainder term, becomes y R = -iJds Z^expd^s) 2^±QL exp(-i^ /s) - O o o 0 +i 2(dt ds 2^ 0exp(i^ 0t) ^ e x p U ^ ) ^ : f i L exp(-i/^( s+t)), ~ ° ° - o o when the order of integration of the double i n t e g r a l I s changed and the time s i s translated to s-t. In Sec. 5 . 3 , the generalized cross section w i l l be written i n terms of the 158. transition superoperators ^ Q and P ^ i s t * This transition superoperator can he put in parameter-ized form when use i s made of the eigenfunctions of » In particular,the distorted transition superoperator acting on an arbitrary statistical" state, S(t',),can be written as l ( y - 5 ^ F v . K i V ) - (5.2.2*) in terms of the three parameter distorted transition super-operator o fr Here the parameterized "internal" transition superoperator i s ( 5 . 2 . 2 6 ) Eq. ( 5 . 2 . 2 5 ) follows from Eqs. ( 5 . 2 . 2 0 , 2 1 , 2 3 ) using indepen-dent convergence factors for each time integral. Snider and Turner''" used the parameterized transition superoperator, y(v-X+iy), defined by 159. i n t h e i r treatment of the double potential case. It was written as the sum ^(y-X+i^) = ^/ Q(vX+i^) + - ^ L (vx+if) o ~~ /-^int^ ' S * 1 ? ^ LQ^-Z (5.2.28) involving the parameterized i n t e r n a l superoperator,Eq. ( 5.2 . 2 6 ) . To derive the operator formulas f o r the quantal cross sections, the ^  parameters i n the d i f f e r e n t parts of the second term i n Eq. (5.2.28) were assumed to be indepen-dent. This assumption i s seen to be j u s t i f i a b l e since the appropriate terms i n the right hand side of Eq. (5.2 .27) must be equal to the right hand side of Eq. (5.2.24). 160. 5.3«CROSS SECTION The generalized cross section with (or without) non zero momentum dispersion involves the transition superoperator %,see Eq. ( 4 . 2 . 4 ) or Eq. ( 3 . 4 . 1 5 ) . In Eq. ( 4 . 2 . 4 ) the eigenfunctions of $C were used as a basis for representing^. This expression can also be written in terms of the phase space representation;namely ° /gen (P"^ R ) " - iIdPP2{drfdr/dP, o where use has been made of the spread function f ?(p'), P Eq. ( 4 . 3 * 2 ) . Using Eq. ( 5 . 2 . 1 6 ) ,the generalized cross section becomes the sum of two terms, (5 .3*2) The f i r s t i s the generalized cross section for the reference system. The second i s duetto the distorted transition super-operator, <-/d^ s^ .,Eq. ( 5 . 2 . 2 0 ) . The distorted part of the cross section involves the position integrals of the phase space representation of "7^^*namely JdrJdr^ r.p«|^ lsJr'(p^  =Jdrjdr^ ,pR (l-ijdt 2^ 0exp(ii t) ( o 161. ( 5 - 3 . 3 ) Eq. ( 5 . 3 . 3 ) results from Eqs. ( 5 . 2 . 2 1 ) and ( 5 . 2 . 2 3 ) when i t is noted that the integral over r' of the ideal phase space element j£'»P'^ ^ s a n eigenket of with zero eigen-frequency, J d r ' e x p C - i ^ s j I r ' . g ^ = J d r ' |r• . g ' ^ . ( 5 - 3 . 4 ) As well, the integral over r of the ideal phase space element <^r,pR| is an eigenbra of again with zero eigenf requency, ^ r , P R | e x p ( - i ^ s ) = Jdr,^r,pR|. (5-3-5) dr' A / . Because Eq. (5*3*3) i s a matrix element of the distorted transition superoperator between an eigenbra and an eigenket having the same frequency (in this case zero),then i t can be written as =Jdrjdr' ^r,pR | ( l - i f d t e x p ( - i ^ t ) ^ e x p ( i t ^ t ) ) y ft = |dr dr" ' ^ • ^ | ^ E 0 ^ i n t i i - L 0 U ' . £ ^ • (5.3. 6) 162. This expression involves the transpose of the reference Miller superoperator S X . = lira exp(-i )exp(i ^o £-»-oo u = 1 -ijdsexp(-i^s) ^ e x p ( i ^ s ) , (5 .3*7) as well as the abstact internal transition superoperator, defined by o r ^ i n t = ^1 " i | d s ^ e x p ( i / s ) ^ . (5*3*8) The distorted part of the generalized cross section i s then j^.p«l A J E 1 -s/j f p " ( £ ' K ( 5 0 , 9 ) Because of the nature of '^n-t»'fc5lis cross section i s the sum of two terms, gen ^(p"^R) = - i |dpp2/dr |dr' dp* and ° /«.!S,(r)(p,,-*r) = -rdsfdpp2/dr/dr,|dp 1 6 3 . ^ N v ( 5 . 3 . 1 1 ) The f i r s t term involves the superoperator "2-^,dressed by the reference M i l l e r superoperator and i t s transpose. This term corresponds to the f i r s t term f o r the single potential as i s seen i f 2^ i s chosen to be zero so that ^ i s then equal to The second term i s i n the form of a time c o r r e l a t i o n function involving the superoperator " 1 , again dressed by the reference M i l l e r superoperator and i t s transpose. The dynamical motion of t h i s time c o r r e l a t i o n function i s generated by the f u l l superoperator o£\ This term i s the generalization of the time c d r r e l a t i o n function term that appears i n the single potential case. The f u l l generalized cross section,Eq. ( 5 » 3 « 2 ) , has been written i n terms of the time i n t e g r a l superoperators here,in contrast to the parameterized superoperators used by Snider and Turner f o r the quantal generalized cross section. The expressions developed here are v a l i d f o r both c l a s s i c a l and quantum mechanics. 164. 5.4.QUANTAL DWBA For the case of a double potential,the exact general-ized cross section has been written i n terms of the reference cross section and a distorted cross section. The distorted part contains the i n t e r n a l t r a n s i t i o n superoperator, -^n-j-i given by Eq. (5»3»8). In the distorted wave Born approx-imation (DWBA), the i n t e r n a l t r a n s i t i o n superoperator, ^ i n t ' ^ " s r e P l a c e c * by the superoperator o Tint - *i - i p s ^ e x p d c ^ s ) ^ , (5.*.1) — OO so that the DWBA cross section becomes the sum of three terms j ^*p,DWBA( - } = ( r ^ p , ( 0 ) ( g } gen  KZ gen v £ + C rgeS' ( l ) (P"- >« ) + °"g!S , ( 2 )(P"-R). ( 5 . 4 . 2 ) The second order i n ^ term involves a time c o r r e l a t i o n function o oo (yg^'(2)(p"-,R) = JdsJdpp2jLjdrjdp/ <^rA£l^ ^ e x P ( i < s ) ^ i 2 L U - , A fj(p.). 1 0 o l W ~ ( 5 . , . 3 ) The equivalence of t h i s approximation of the superoperator formulas to the usual quantal operator expressions f o r the DWBA cross section i s now shown. Here,the algebra involved 165. requires no assumptions i n contrast to the parameterized i version used by Snider and Turner . To perform the algebra,the d e f i n i t i o n s of the i d e a l phase space elements,Eqs. (2.2.28) and (2.2.29).are required. As well,the action of the reference M i l l e r superoperator on an operator (say A) i s required. That product i s si A = n^n , (5.kA) o where the r i g h t hand side involves the reference M i l l e r operator, Q = lim exp(iH Qt/h)exp(-iKt/n) t •*-o© o = 1 -in~ 1 -[dsexp(iH 0s/n)V ( )exp(-iKs/n) 9 = 1 -ih " 1 J >dsexp(iKs/n)t 0exp(-KsA). (5.^.5) -Co and i t s adjoint. The l a s t form of Eq. (5.4.5) involves the reference t r a n s i t i o n operator t As well,the transpose of the Moller superoperator s a t i s f i e s an equation s i m i l a r to Eq. (5.4.4) -ft-L A= i l 0 A ( I X 0 ) , (5.^.6) o where SLQ i s the transpose of the reference Moller operator. The p o s i t i o n i n t e g r a l s of the phase space representation 166, of the f i r s t term in Eq. (5.4.l) become = h - 3 T r J d r J d r ' JdPjdP'expCir-P/n) |PR-iP>< pR+^PI - ^ L |E '-*r><E , +*Elexp(ir"-g'/h) o ssl - h 3 T r |p6><pR|XlL ^ I 1 L IJ'XP'I 0 t = hV 1< :<(pR)iv1|(pj')+/><'piso|p«) .( PR |s 0|p 1X(g ,) + |v1|(pR)"^> ), ( 5 . * . 7 ) in terms of the incoming eigenfunctions l(p')+> = -ft-cU*) (5.^.8) and the outgoing eigenfunctions, |(pR)->= ( A 0 ) \ R > . 167. (5.^.9) of the reference hamiltonian This expression also involves the reference scattering operator S Q = SL^Si^, Using Eq. (5«^ •'*>")• and a similar expression for the transpose of the reference Miller operator, the reference scattering operator can be written i n terms of the reference transition operator, CO = 1 -in'^dsexpCiKs/hJtQexpC-iKs/n). (5.^.10) -oo The matrix element of the; reference scattering operator involved in Eq. (5»4.7) i s then ^ P R | S 0 I P « ) = $(pR-p/) -i^tt/p)(2tr)S(p-p-') <p 'R|t 0\p'> . (5.^.11) As well,the complex conjugate of this matrix element i s also required. Using Eq. (5»4.1l),Eq«, (5«407) becomes = Zih 3*. - 1 S(pR-p,)Im<(pT|v1|(p')+> 168. + 2i^/p)(2rr)o :(p-p')h 3 r : lRe<(p'R)-|v il(p') +> < p'j tQ\p'R> . (5.4.12) In the limit of zero momentum dispersion,the contribution to the generalized cross section from the f i r s t order i n term i n Eq. (5.3.10) i s then C T g e£' ( l )(p"-*R) = 2(2tr) 2h 2 y a 2Re<(p"R ) - | v i|(p") + <> < p" |tg| p"R> + 4Th2(/*/p,l)S(R-P" )Im<(p" )"| \{g )+>, (5.4.13) which i s recognized as the f i r s t order i n V part of the generalized DWBA. cross section,see e.g. Snider and Turner . This term contains both loss and gain contributions to the generalized cross section. The second term i n Eq. (5.4.1) when used in Eq'. (5.3.11) gives the second order i n contribution to the DWBA generalized cross section,Eq. (5.4.3). In the parameter-1 ized form used by Snider and Turner ,the reduction of this term to the standard operator expressions required an assumption about the dependence i n the limit 0 + . In the present time integral formalism,no such assumption is. necessary. As with the f i r s t order term,the second order term involves the position integrals of the phase space represen-tation of this term, 1 6 9 . -(2Th) 2( /c / p ' ) J ( p - p ' ) | < ( p'R ) - l v il ( p ; ' ) + > | 2 where no assumptions have been made. Using Eq. ( 5 . 4 . 1 1 ) , the second order contribution to the DWBA generalized cross section,in the limit of zero momentum dispersion,becomes Again,'.this i s recognized as the second order i n contri-bution to the generalized DWBA cross section,see e.g. Snider and Turner . The DWBA generalized cross section has been developed from a time integral superoperator viewpoint. The reduction to the standard operator formulas has been achieved without making any assumptions in contrast to the treatment of Snider and Turner which involved parameterized superoperators. ( 5 . 4 . 1 5 ) 170. 5.5.CLASSICAL CROSS SECTION AND CLASSICAL DWBA  5.5.a.EXACT CLASSICAL CROSS SECTION The exact classical cross section i s given by Eq. (5 .3*2) . In particular,the distorted part of this cross section, Eq.. (5»3«9).involves the phase space representation of the superoperator product XI L ^ CL ^ int,CL L ,CL* T h i s contains the phase space representation of the reference Miller superoperator and of i t s transpose,which can be written i n terms of displacements,!see for example Eq; (2.6.11 ),^  or which can be written i n terms of the reference motion subject to asymptotic conditions. Here,it i s convenient to write the reference Miller superoperator in terms of the displacements while i t s transpose i s written in terms of the asymptotic conditions. In particular,Eq. (5«3«9) involves the product of XI ^ Q-^ and the spread function f »(p'), which i n the phase space representation i s (5.5.1) The vector A/.,. (r'l',p'") i s the displacement of the momentum ~ p ~ ti ^ due to the reference motion,see Eq. (2 .6 .13) . Eq. (5«3»9) also involves the product of the ideal phase space element r,pRJ and the transpose of the reference Miller super-operator, 171, <fr, PR ( I l L C L o = lim ^r,pR\exp(i^(t )exp(-iofn P T t ) = lim ^r+pRt^M.,pR|exp(-io^0 C L t ) ±•900 jftV* ' = ^ R o ( 0 l £ » p R ) » P o ( 0 l r ' P g ) l » (5.5.2) where the position R*(o\r,pR) and the momentum P^Olr.pR) satisfy Hamilton's equations of motion for the reference system subject to asymptotic conditions (at large positive times) 11m (RQ(0lr,p3) - (r+pRt/w.)) = 0, ( 5 . 5 . 3 ) and lim (P*(0|rf<pR) - pR) = 0. (5*5 .4) t*co u " Physically,the phase space point (r,pR) i s taken forward along the free path to the asymptotic future.and then the motion i s brought backward along the reference trajectory for an equal time period to the phase point (R*( 0lr,pl^), Pj ( 0|r/ PR)). The distorted part of the exact classical cross section for double potentials,Eq. (5.3«9)»is then t r g ^ d i s t ' C L ( p » - » 8 > = -ijdpp 2jdr|dr'jdp' 172. ^ ( P - A ^ r ' . p / ) ) . ( 5 . 5 . 5 ) Aj AS This expression involves a matrix element of the internal transition superoperator s~7^n^ C L which i s defined by Eq. ( 5 » 3 . 8 ) . Since i t consists of two terms,then the cross section Eq. (5»5«5) i s also the sum of two terms. One of these terms i s f i r s t order in ¥ ,namely o o . j — — r r * P ( p ; ( o » r.p8 ) + d ? j ( p ; . i , ; ) ) . ( 5 . 5 . 6 > A j (J AS + + A ( 0 ) + + where the variables RQ and jgQ in A p+ (RQ 'VO^ A R E E ^ U A L ' T O R0(0|r,pR) and £ Q(01r,pR),respectively. The displacement Ap+ (gJ.Pj),given by Eq. ( 2 . 6 . 1 3 ) , i s " 0 " ~ ( 5 . 5 . 7 ) i n terms of the position R n(t|r,pR). It i s depicted i n Fig. 5 . 1 . Physically,the trajectory that goes asymptotically as A A , r+pRt/w and pR for large positive times,is traced backward a l l the way to large negative times where i t i s again free. The resulting free motion i s brought forward to zero time to define A.^®) and A 2^^ » I n fact,this i s just what the AJ -n T ~j T 3 ~ ~o - 0 173. F i g . 5 . 1 . T h e T r a j e c t o r y R Q ^ I ^ ' P * * ) 174 classical reference scattering superoperator, ^O.CL = ,CL-^L ,CL» (5-5O8) ' o o does. Eq. (5«5'»6) can be written in terms of C L when i t i s noted that the spread function becomes =^*ids' (,'CfioCo,?'pR,'?o(oir'P« ,l-^Lo,i P ^ In terms of -?x0 Q-^the f i r s t order contribution to the cross section i s then < ) P where J (p"->R) = J d p p 2 d r d r ' d p ' x * ~° £ *> i v J ~ * Z ^ R j ( 0 | r , p R ) ^ ( 0 . | r , p R ) ^^(Ojr .pS) ^ 1 = ^ ' P ^ L o , G L ^ . C L ^ . C l ) ' (5 .5 .11) 1 7 5 . Inserting Eq. ( 5 . 5 . 1 1 ) into Eq. ( 5 . 5 . 1 0 ) and using the identity ( H L f C L >n L , C L = P 0,ci/ ^- 5- 1 2 ) 0 o ' where (PQ ^  i s the projection on to the unbound classical motion,reproduces the starting point,Eq, ( 5 « 3 . 9 ) « The remaining term in the distorted part of the classical cross section i s of the form of a time correlation function,Eq. (5«3«ll)» This time correlation function involves the superoperator product X*p*l^ L 0 ' € L ^ • C L 6 X P ( 1 ^ C L S > ^ 1 , G L I 5 ' ^ % (R*(0lr,<pR)) } /,+ A <*R>lr,pR) ^ P 0 ( 0 l r , p R ) -P+( 0\ r, pR)| exp( i ^ L s ) ^  , C L |r',p ^  ) 4 S(P+(s)-p')£(R+(s)-r') ), ( 5 . 5 . 1 3 ) P +(s) where the position R +(s) and the momentum P +(s) satisfy Hamilton's equations for the complete system subject to the i n i t i a l conditions R + ( 0 ) = R^Olr^pR) and P + ( 0 ) = P Q ( 0 j r f p R h Eq. (5*3*ll)»for the classical case,is then 176. (R) CL A f 7 2 P V iCR^olr .pS) °*geS'( }' C L(£"^ R) - Jdajdpp 2/dr 1 W ° - • i R + 0 ( 0 | r . P R ) r ^ ( O l r . p R ) <K ( 5 + ( s ) ) ) I. ^ R + ( s ) i p + ( s ) f n P ( P (s) P « +~V(s)<5 (3),P+(S))J . (5.5-14) while the exact c l a s s i c a l cross ( 5 . 5 . 6 or .10),(5.5.14) and the section. section i s the sum of Eqs. c l a s s i c a l reference cross 1 7 7 . 5 . 5.b.REFERENCE TRAJECTORY APPROXIMATION: THE CLASSICAL DWBA The distorted part of the exact c l a s s i c a l generalized cross section involves the t r a j e c t o r i e s s R +(s) = R * ( 0 l r v p R ) + j d t P + ( t ) / ^ c , ( 5 . 5 - 1 5 ) and + + . * f ^ V ( R + ( t ) ) P (s) = P 0(o|r,«PR) J^dt - y - r — ( 5 - 5 . 1 6 ) <)R +(t) generated by the complete c l a s s i c a l L i o u v i l l e superoperator o f T h e DWBA was defined by replacing the complete dynamics with the reference dynamics i n the i n t e r n a l t r a n s i t i o n superoperator,see Eq. ( 5 . 4 . 1 ) . In the quantal case,this approximation was shown to be equal to the usual DWBA t r a n s i t i o n operator expression f o r the cross section,Eqs. ( 5 . 4 . 1 3 ) and ( 5 . 4 . 1 5 ) . Since Eq. ( 5 . 4 . 1 ) i s also v a l i d i n c l a s s i c a l mechanics,it defines a c l a s s i c a l DWBA generalized cross section. I t i s equal to replacing the complete t r a j e c t o r i e s R +(s) and P +(s) with the reference t r a j e c t o r i e s R^tsJr^pR) and P^slr^pS). In t h i s approximation/the spread function i n the time c o r r e l a t i o n function can be written i n terms of the c l a s s i c a l reference scattering superoperator, S ) £'<P*>*~ <i s) (J?0< s > ^ 0 ( 3 ) } Z ~u ^ ^ ' ^ ^ ' ^ L 0 . G L e x P ( I ^ 0 , C L s ) ^ L Q,CL 178. To obtain Eq. (5.5.17),Eq. (5.3.4) has been used along with the intertwining r e l a t i o n s h i p ' o * L o,CL (5.5.18) The c l a s s i c a l DWBA generalized cross section i s then o- ^ p . D W B A » C L ( p » ^ R ) = < r r P ' , ( 0 ) » C L ( p » ^ R ) gen gen pR) pR) dt U ( 2 ) b s Q ( t ) (5.5.19) i n terms of the outgoing free s t a t i s t i c a l state s Q ( t ) which i s defined by » 0 ( t ) m * 0 . C L ' i . M ' (5.5.20) where s-(t) i s the original incoming s t a t i s t i c a l state 179. In contrast to the c l a s s i c a l Born approximation,the c l a s s i c a l DWBA has a value f o r a l l observable angles R due to the presence of the c l a s s i c a l reference scattering superoperator. 180. 5.6.STATIC AND DYNAMIC INTERFERENCE APPROXIMATIONS For the case of a double potential(whether c l a s s i c a l or quantal),the exact generalized cross section has been written as the sum of the reference cross section and a distorted cross section,Eq. (5«3«2). The distorted part contains a matrix element of the i n t e r n a l t r a n s i t i o n superoperator and thus involves the sum of two terms,one being the superoperator 2 ^ ,the other being a time i n t e g r a l of the superoperator product ^ e x p ( i c i f s ) Z .^ This time c o r r e l a t i o n function i n the second term i s s i m i l a r to the time c o r r e l a t i o n function f o r the case of a single potential, Eq. (4.2^6),and,thus i s i n a form f o r making s e m i c l a s s i c a l -type approximations s i m i l a r to the DIA and the SIA. There are two places where differences between quantal and c l a s s i c a l mechanics a r i s e i n the superoperator product XL L - ^ i n t ^ L » n a m e l y i n s t a t i c - l i k e differences o o and i n dynamic-like differences. This s i t u a t i o n leads n a t u r a l l y to two d i f f e r e n t semiclassical-type approximations, a dynamic interference approximation (DDIA) and a s t a t i c interference approximation (DSIA),for double potentials. In the DDIA,the quantal s t a t i c interference e f f e c t s i n the superoperators Ii L tP and i l L are ignored. Thus, o o these quantal superoperators are replaced by t h e i r c l a s s i c a l counterparts, CL' CL a n d CL* T h e <l uantal o' ' o' dynamical interference e f f e c t s are retained i n t h i s approx-imation through the use of the quantal generator °^Q« For definiteness,since the reference M i l l e r superoperator (and 181 i t s transpose) are taken as c l a s s i c a l , t h e reference cross section i s also taken as c l a s s i c a l . The generalized DDIA cross section i s then + O- r P ' d i s t ' D D I A ( p » - , R ) ( (5.6.1) gen %, where the distorted part of the DDIA i s Oo n 0 - g J p . d i . t . B D I A ( g . ^ S , . .^fefe-fa- ^ r . p f i l - ^ " L ,CL( Z / l T C L - i J d s 2 f l . C L E X P ( i ^ Q s ) ^ , C L ) J ^ L , CL ( 5 . 6 . 2 ) Replacing the f u l l quantal generator by the reference quantal generator Q S^ V E S ' t i i e corresponding DWBA to the DDIA. Further evaluation of the f u l l (or DWBA) DDIA requires knowledge of the exponential of the generator ^ (or citQ Q). These approximations are not pursued here. In the DSIA,the quantal dynamical interference ? effe c t s are ignored by replacing the quantal generator O^Q by i t s c l a s s i c a l counterpart «^ c i,while the quantal s t a t i c interference e f f e c t s i n the superoperators L Q , y 2h n and .0. T n are retained. For definiteness,the reference cross section i s taken as quantal sinee ;the reference 182. M i l l e r superoperator (and i t s transpose) are taken quantally. The generalized DSIA cross section i s then cy * p « D S I A ( P » + R ) = <r ^ 0 ) - Q ( P - - » R ) v gen v £ gen it + < r * p . d i a t . M I A ( s . ^ 8 ) > ( 5 > 6 o ) w&ere the distorted part of the DSIA i s ^ p R | H L o i Q ( ^ -ijds ^ ( Q e x p ( i < L s ) i * f As with the DDIA^replacement of the c l a s s i c a l generator °^CL ^ r e f e r e n c e generator OX^Q leads to the corres-ponding DWBA to the DSIA. Further evaluation of the DSIA requires knowledge of the quantal reference M i l l e r super-operator and i t s transpose. This approximation i s not pursued here. Rather,in the next section, a c l a s s i c a l t r a j e c t o r y approximation (DCTA) f o r double potentials i s defined where a l l the ef f e c t s of the reference pote n t i a l are taken as c l a s s i c a l . This approximation i s pursued i n greater d e t a i l . 183. 5.7.CLASSICAL TRAJECTORY APPROXIMATION In the single potential case, the SIA and the CTA are i d e n t i c a l . Por the case of double potentials,the DCTA i s defined as an approximation to the DSIA. That i s , i n order to evaluate the DSIA,some knowledge of the quantal reference. ffyzfller superoperator and i t s transpose i s required. The DCTA i s defined by taking a l l the effects of the reference system as c l a s s i c a l , i n c l u d i n g the reference M i l l e r superoperator,leaving only the s t a t i c interference e f f e c t s inherent i n Q . The DCTA generalized cross section i s then *gen " D C T V - "> " ^ ' ( 0 ) , ° L ( P ^ R ) + „, «p.diat.DOTA ( l^, i ( 5 . 7 > 1 ) gen «v where the distorted part of the DCTA i s o ~ o —CO ( 5 . 7 . 2 ) The c l a s s i c a l reference M i l l e r superoperator and i t s transpose have been written i n terms of the spread function 184. and the i d e a l observable phase space element as i n Eqs. (5.5.1) and ( 5 . 5 . 2 ) . The f i r s t order i n ^ contribution to the DCTA cross section i s simply ^ ' ( l ) ' D C T V ' - > £ ) = ^n-^dpp^dX'fdp' ^ 1(2(p ,-PQ(0|r,p^)))sin(2RQ(0|r,pR)-(PQ(0|r,pR) -p')/h)A(pi + A|.?)(JR;,p')). ( 5 . 7 . 3 ) where Eq. ( 2 . 3 . 2 5 ) has been used f o r the phase space representation of 2^ Q . Eq. (5«7»3) i s comparable to Eq. ( 5 . 5 . 6 ) which i s the f i r s t order c l a s s i c a l term. Indeed, i n the l i m i t h-*0,Eq. (5 .7*3) "becomes Eq. ( 5 . 5 . 6 ) as the "h ef f e c t s have been l e f t i n the phase spase representation of 2^ Q . The major difference between Eqs. (5«5«6) and ( 5 . 7 . 3 ) i s due to the non-locality i n the momentum dependence of the phase spase representation of <^ Q . I t was the l o c a l i t y i n both the position and the momentum of ^ that allowed the f i r s t order c l a s s i c a l term to be simply written i n terms of V Q C L , s e e Eqs. (5.5.10) and (5«5«ll)« Eq. ( 5 . 7 . 3 ) can be written i n terms of Q Qjj,but the 2/"^ ^  i s then sandwiched between the transpose of the reference M011er superoperator and i t s adjoint,as i n Eq, ( 5 . 5 .11). bince n i s not l o c a l i n the momentum,then the product SLjj C L ^ 1 Q ^ - ^ - j , ciP * s n o n""l°cal and then the 185. representation of -ff ^ C L i s not diagonal i n terms of the observable d i r e c t i o n R. The remaining term i n the DCTA cross section has the form of a time c o r r e l a t i o n function involving the super-operator product I ^ ^ V i r ' P ^ ^ L o ' C L ^ ' Q e X P ( i ^ C L S ) ^ = -2 8r 2[dp[dP« V 1(2(P-Pj ( 0)r,pR)))V 1(2(P'-P(s))) sin(2Rj ( 0|r,pR)'(PQ ( 0 l r,pR)-P ) /K ) s i n ( 2 R ( s ) ' ( P ( s ) - P , ) / n ) f * p ( P ' + A^? )(R(s);P')). ( 5 . 7 . 4 ) Eq. ( 5 . 7 . 4 ) involves the position R(s) and the momentum P(s) which s a t i s f y Hamilton's equations of motion f o r the complete system,generated by o^,^, subject to the i n i t i a l conditions R(0) = R^Olr.pR) and P(0) = P. The remaining part of the DCTA cross section i s then C - J P ^ R ) ' D C T A ( P » - R ) = 2 % - 2 / d s ] l p p 2 | d r j d P ^ P ' V1(2(P-PQ(01r,pR)))V±(2(P'-P(s)))sin(2RJ(0}r,pR) ' (P^ ( 0|r,pR)-P)/n)sin(2R(s)-(P(s)-P')/n) fJiP' + A^hRis),^)). . ( 5 . 7 . 5 ) 186. Eq. ( 5 » 7 » 5 ) becomes the c l a s s i c a l r e s u l t Eq. (5.5.14) when the appropriate and sine terms are combined i n the h-» 0 l i m i t . As with the single p o t e n t i a l time c o r r e l a t i o n function,this can be written as the sum of two terms i f the product of the sines i s written as the difference of cosines. The DWBA to the DCTA res u l t s when the pos i t i o n R(s) and the momentum P(s) are replaced by R n(s) and ^ ( s ^ t h e appropriate p o s i t i o n and momentum generated by the reference motion. 1 8 7 . 5 . 8 . D I S C U S S I O N T h e c o l l i s i o n p r o c e s s h a s b e e n d i s c u s s e d f o r d o u b l e p o t e n t i a l s . T h e g e n e r a l i z e d c r o s s s e c t i o n w a s w r i t t e n a s t h e s u m o f t h e r e f e r e n c e c r o s s s e c t i o n a n d a d i s t o r t e d c r o s s s e c t i o n . T o d o s o , t h e t r a n s i t i o n s u p e r o p e r a t o r w a s w r i t t e n a s t h e s u m o f t i m e i n t e g r a l f o r m s o f t h e r e f e r e n c e t r a n s i t i o n s u p e r o p e r a t o r a n d o f a d i s t o r t e d t r a n s i t i o n s u p e r o p e r a t o r , i n c o n t r a s t t o t h e w o r k o f S n i d e r a n d T u r n e r 1 w h o u s e d p a r a m e t e r i z e d s u p e r o p e r a t o r s . A s w e l l , t h e c r o s s s e c t i o n d e f i n e d i n t h i s c h a p t e r i s v a l i d f o r b o t h c l a s s i c a l a n d q u a n t u m m e c h a n i c s . F o r t h e q u a n t a l c a s e , S n i d e r a n d T u r n e r h a v e s h o w n t h a t a p p r o x i m a t i n g t h e r e s o l v e n t f o r t h e c o m p l e t e v o n N e u m a n n s u p e r o p e r a t o r ex?f w i t h t h e r e f e r e n c e r e s o l v e n t g i v e s t h e g e n e r a l i z e d q u a n t a l D W B A c r o s s s e c t i o n . T h i s r e d u c t i o n o f t h e s u p e r o p e r a t o r f o r m u l a s t o t h e s t a n d a r d o p e r a t o r f o r m u l a s r e q u i r e d a n a s s u m p t i o n a b o u t t h e p a r a m e t e r -i z a t i o n o f t h e d i s t o r t e d t r a n s i t i o n s u p e r o p e r a t o r . T h i s r e d u c t i o n w a s r e p e a t e d i n S e c . 4 u s i n g t h e t i m e i n t e g r a l ; f o r m a l i s m , i n w h i c h n o a s s u m p t i o n s w e r e r e q u i r e d . T h e e x a c t c l a s s i c a l d o u b l e p o t e n t i a l c r o s s s e c t i o n w a s c o n s i d e r e d i n S e c . 5» I t w a s w r i t t e n i n t e r m s o f t h e c l a s s i c a l t r a j e c t o r i e s f o r t h e c o m p l e t e m o t i o n . R e p l a c i n g t h e c o m p l e t e t r a j e c t o r i e s b y t h e r e f e r e n c e t r a j e c t o r i e s d e f i n e d t h e c l a s s i c a l g e n e r a l i z e d D W B A c r o s s s e c t i o n . W h i l e t h e c l a s s i c a l B o r n a p p r o x i m a t i o n i s e s s e n t i a l l y a f o r w a r d c r o s s s e c t i o n , t h e c l a s s i c a l D W B A h a s a v a l u e f o r a l l 188. observable directions due to the reference motion. Since the exact quantal distorted cross section involves a time c o r r e l a t i o n function,then i t i s i n the same form f o r making semiclassical-type approximations as was the single p o t e n t i a l cross section. Three d i f f e r e n t semiclassical-type approximations have been presented. The DDIA ignored the s t a t i c - l i k e quantal interference ef f e c t s while r e t a i n i n g the f u l l quantal dynamics generated by O^ Q» The DSIA ignored the dynamical quantum ef f e c t s by using the exact c l a s s i c a l motion generated by ° ^ G L while ret a i n i n g a l l the s t a t i c - l i k e quantal interferences. These approximations were not pursued here. The t h i r d semiclassical-type approximation,the DCTA,retained only the s t a t i c interferences i n 2 ^ ^ while taking a l l other superoperators as c l a s s i c a l . I t was written i n terms of the c l a s s i c a l t r a j e c t o r i e s . CHAPTER 6 KINETIC CROSS SECTIONS:TRAJECTORY APPROXIMATIONS 190 6.1.INTRODUCTION Binary c o l l i s i o n processes can be described i n terras of the t r a n s i t i o n superoperator, 7J1. In p a r t i c u l a r s generalized cross section i s the expectation value of a f l u x observable i n the s t a t i s t i c a l state that arises by binary interactions from a state that was asymptotically free i n the distant past. This leads to a v a r i e t y of expressions f o r the generalized d i f f e r e n t i a l cross section,see f o r example Eqs. (3*4.15) and ( 4 . 2 . 4 ) . Depending upon the p a r t i c u l a r i n i t i a l free s t a t i s t i c a l state and the f l u x observable i n question,many d i f f e r e n t cross sections can be J 2 defined. Of p a r t i c u l a r i n t e r e s t are ki n e t i c cross sections * T 4 J* where the i n i t i a l s t a t i s t i c a l state includes a Boltzmann weight. The act i o n of the t r a n s i t i o n superoperator on t h i s s t a t i s t i c a l state produces the in t e r a c t i n g s t a t i s t i c a l state f o r binary t r a n s i t i o n s . P a r t i c u l a r k i n e t i c cross sections correspond to the expectation values of d i f f e r e n t observables. In the present work,internal state k i n e t i c cross sections, defined i n Sec. 6 . 2 ,are emphasized as opposed to v e l o c i t y dependent cross sections . Only on-the-frequency s h e l l cross sections are investigated i n the remainder of t h i s thesis. These cross sections can be (and are) written i n a symmetric form 2. In Sec. 6 . 3 .the symmetric i n t e r n a l k i n e t i c cross section i s written i n terms of the double potential formalism of chapter 5« In t h i s case,the reference p o t e n t i a l i s assumed not to cause t r a n s i t i o n s between i n t e r n a l states. 191 The r e s u l t i n g cross section i s then a matrix element of a frequency parameterized i n t e r n a l t r a n s i t i o n superoperator, and thus involves a time c o r r e l a t i o n function. As with the two previous chapters,various semiclassical-type approx-mations may be considered. The time c o r r e l a t i o n function i s f o r quantities B. . involving the action of the super-operator 2^ on the product of an i n t e r n a l state coherence and a Boltzmann weight associated with the reference Hamiltonian. In Sec. 6 . 4 ,a semiclassical-type approximation SCL to the t r a n s l a t i o n a l part of the quantity B^^,termed B ^ , i s defined by taking as quantal while the Boltzmann fa c t o r i s calculated c l a s s i c a l l y . As well,a c l a s s i c a l CL approximation,B..,is considered where a l l the t r a n s l a t i o n a l J e f f e c t s are treated as c l a s s i c a l . In both cases,the i n t e r n a l states are treated quantally. In Sec. 6.5»the f u l l quantal distorted wave Born approximation (DWBA) i s considered as well as two types of semiclassical approximations to the DWBA. One approximation, the k i n e t i c d i s t o r t e d wave dynamic interference approx-imation (KDWDIA),is defined by retaining the dynamic-like interference e f f e c t s i n the t r a n s l a t i o n a l part of the exponential of the reference generator while ignoring the CL s t a t i c - l i k e interferences by using B... As with previous DIA-type approximations,the KDWDIA i s not pursued here. On the other hand,the k i n e t i c distorted wave s t a t i c interference approximation (KDWSIA) i s defined by retaining the f u l l quantal nature of the B-^'s while taking the t r a n s l a t i o n a l 1 9 2 part of the exponential of the reference generator as cl a s s -i c a l . Taking the Boltzmann weights as c l a s s i c a l . t h a t i s , replacing B. . by B?9 L i n the KDWSIA defines the k i n e t i c distorted wave c l a s s i c a l t r a j e c t o r y approximation (KDWCTA). This s t a t i c - l i k e approximation scheme i s completed by defining the k i n e t i c distorted wave f u l l c l a s s i c a l approx-imation (KDWFCA) where only the i n t e r n a l states are consid-ered quantally. In Sec. 6.6,the KDWCTA and the KDWFCA are written i n terms of the c l a s s i c a l reference t r a j e c t o r i e s . As well, straight l i n e approximations are presented. In the f u l l c l a s s i c a l case,the resultimg straight l i n e cross section i s equal to the "constant acceleration approximation" (CAA) of Oppenheira and Bloom . Hynes and Deutch have previously shown t h i s to be equivalent to a straight l i n e approximation. The straight l i n e approximation to the KDWCTA then represents a p a r t i c u l a r semiclassical extention to the CAA. 193 6.2.KINETIC CROSS SECTIONS Generalized t r a n s i t i o n rates due to binary c o l l i s i o n s are proportional to the t r a n s i t i o n superoperator*. In p a r t i c u l a r , f o r a s t a t i s t i c a l state S,the quantity i ^ S i s a measure of the rate of a l l t r a n s i t i o n s from the s t a t i s t i -c a l state S. The expectation value of an observable A with the quantity i ^ S playing the role of the s t a t i s t i c a l state, i s then a measure of the rate at which A i s obtained by binary c o l l i s i o n a l t r a n s i t i o n s from the s t a t i s t i c a l state S. If A and S involve the se l e c t i o n of p a r t i c u l a r momentum directions,then the generalized d i f f e r e n t i a l cross sections 2 3 4 of chapter 3 are obtained. Here,kinetic cross sections ' ' are obtained by requiring the observable A to be a unit 2 3 4 observable while the s t a t i s t i c a l state S i s usually »^» taken as the product of an i n t e r n a l state term S^ n^ and a Maxwellian momentum d i s t r i b u t i o n h3(27T/ft. kT)~3//2exp(-y<S K t r ) at temperature T ( ^ s = ( k T ) ). The k i n e t i c cross section i s then defined as <9(A|S) = ^ S 1 ^ l 1 ^ ! ^ • (6.2 .1) where (y)^ i s the average speed of the p a r t i c l e s i n the s t a t i s t i c a l state S. In the phase space representation,the Maxwellian momentum d i s t r i b u t i o n i s given by f( r , p ) = h 3 ( 2 T / k T ) - 3 / ^ p | e x p ( ^ K t r ) ^ = (2T/AkT)* 3/ 2exp ( - p p 2 /2yK/). 194. ( 6 . 2 . 2 ) both c l a s s i c a l l y and quantally. I t i s independent of the posi t i o n r.and thus i s not a normalizable s t a t i s t i c a l state. The average v e l o c i t y magnitude over the state S i s then taken as a t r a n s l a t i o n a l average over p only jthat is,the average v e l o c i t y i s ^v> s = (2TyA.kT)~ 3/ 2Jdp(p/^w )exp(- p p 2 / 2 y ^ ) = (8kT/iy*)^ f ( 6 . 2 . 3 ) so that the k i n e t i c cross section f o r t h i s t r a n s l a t i o n a l s t a t i s t i c a l state i s <§r(A)s.nt) . ( i f V / S ^ A l ^ l S l n t e x p ( - p K t r ) ^ . ( 6 . 2 . 4 ) In the phase space representation,the YiJ vanishes,see Eq. ( 6 . 2 . 2 ) , s o that t h i s cross section i s applicable to both c l a s s i c a l and quantum mechanics. P a r t i c u l a r choices of the observable A and the i n t e r n a l state term ^ lead to di f f e r e n t k i n e t i c cross sections. For example,'velocity dependent cross sections* r e s u l t i f the observable A i s the product of an i n t e r n a l part and a t r a n s l a t i o n a l part,see, f o r example,Coombe et a l . As well,the t r a n s l a t i o n a l part of S i s often perturbed from the Maxwellian form. These cross sections are not investigated further here. Rather, 195 only i n t e r n a l state t r a n s i t i o n s are considered. The observable A i s thus taken as the product of an i n t e r n a l state coherence and a Boltzmann weight, A = ja> exp( f ( E a + E b ) A ) <b| , ( 6 . 2 . 5 ) while the i n t e r n a l state i s also of the same formj namely S i n t = ' c> e x p ( " ( 9 ( E c + E d ) / ^ ) <dl' ( 6 . 2 . 6 ) Here,the energy associated with the i n t e r n a l state \ i s E.. The r e s u l t i n g k i n e t i c cross section f o r i n t e r n a l state J changes i s defined as ©'( a b j c d ) = ( i p 2 h 3 / 8 y O ^ | a > < b ) | ^ J | |c> exp(- | 5 K t r ) < d | ^ e x p ( - ^ ( 6 ) c a + 6 ) d b ) A ) . ( 6 . 2 . 7 ) In general,the frequencies of the observable,^ a l 3» and of the state,<^ c d, are d i f f e r e n t . Por the p a r t i c u l a r case 2 that £J a b=d) c d,Coombe et a l have shown that the k i n e t i c cross section,Eq. ( 6 . 2 . 7)»is equal to the symmetric cross section <§) S(ab|cd) = ( i f l 2 h 3 / 8 y . ) ^ a > e x p ( - i p t r ) < b | ^ 7|jc>exp(-^K t r)<d|^ . ( 6 . 2 . 8 ) 196. The d e f i n i t i o n of the k i n e t i c cross section,Eq. ( 6 . 2 . 7 ) , i s 2 consistent with that of Coombe et a l but d i f f e r s from that 7 of Snider and Turner' where the Boltzmann weight was taken as the exponential of fo(^0SL+Ci^)/k instead of, the negative as done here. The equality of Eqs. ( 6 . 2 . 7 ) and ( 6 . 2 . 8 ) holds only on-the-frequency s h e l l ^ a D = ^ c ^ ) » I n the remainder of t h i s chapter and i n chapter 7 fonly the on-the-frequency s h e l l case i s considered,where the symmeterized form of the k i n e t i c cross section,Eq. ( 6 . 2 . 8 ) , i s used. 197 6.3.DOUBLE POTENTIAL FORMALISM In chapter 5 ."the t r a n s i t i o n superoperator was written as the sum of a reference t r a n s i t i o n superoperator and a distorted t r a n s i t i o n superoperator,Eq. (5 . 2 . 1 6 ) . This double potential formalism i s now applied to the symmetric k i n e t i c cross section,Eq. ( 6 . 2 . 8 ) . Furthermore,the reference poten t i a l V Q i s assumed to cause no tra n s i t i o n s between i n t e r n a l states. Because of t h i s r e s t r i c t i o n , t h e term due to the reference t r a n s i t i o n superoperator i s i d e n t i c a l l y zero,while the distorted t r a n s i t i o n superoperator leads to the expression (§)S(ab|cd) = ( i p 2 h 3 / 8 - r r ^ ) ^ i a > e x p ( - i p K t r ) < b l \ ^ L ^ i n t ^ c d ^ L ^ 0 ) ^ - ^ S r K ^ . ( 6 . 3 . 1 ) Eq. ( 6 . 3 . 1 ) i s obtained when ^ 7 ^ i s t # E ( l * ( 5 * 2 . 2 0 ) , i s used i n Eq. ( 6 . 2 . 8 ) which i s an on-the-frequency s h e l l expression. The transpose of the M i l l e r superoperator arises only because the frequencies are equal as was the. case f o r the generalized d i f f e r e n t i a l cross section i n Sec. 5 .3* Here the frequency dependent i n t e r n a l state t r a n s i t i o n superoperator i s o ^ i n t ^ c d * " ^ - i | i s ^ e x p ( - i ( ^ d - ^ ) s ) ^ . ( 6 . 3 . 2 ) The quantal reference M i l l e r superoperator acting on the product of two operators,say A and B,is the product 198 of the reference M i l l e r superoperator acting on A times the reference M i l l e r superoperator acting on B , ILL (AB) =1)^11^ o = ( j f l A ) ( i l L B). ( 6 . 3 - 3 ) 0 o This involves the i d e n t i t i e s ( f o r an a r b i t r a r y operator C) i l C. ^ 0 O i l g , ( 6 . 3 . * ) o and ( 6 . 3 . 5 ) The i d e n t i t y i ^ O ^ e = P t r . c " ® ( H t r > ( 6 - 3 - 6 ) i s also of use. Here,P. i s the projection onto the continuum states (having positive energy l e v e l s ) of the reference t r a n s l a t i o n a l hamiltonian H - t r = K ^ r + v o * I n t h e ^• a s " t form of Eq. ( 6 . 3 * 6 ) , t h i s projection i s given i n terms of the Heaviside function ® ( x ) = | \ x> 0 . ( 6 . 3 . 7 ) 0 x <0 199. The reference M i l l e r superoperator acting on the s t a t i s t i c a l state le}exp(-i p )<d( i s then SX L |e><dlexp(-ifK t ) o - ( f t L \c><d\)<rLL exp(-*jSK. )) o b ' = |c><d|exp(-i|3H t r)®(H t r), (6.3.8) where use has been made of Eqs. (6.3.3).(6.3.6) and the intertwining r e l a t i o n ^Vtr = H t A ' ( 6'3-9) The k i n e t i c cross section Eq. (6.3.1) can then be written as @ S ( a b l c d ) = ( i p 2 h 3 / 8 T ^ ) ^ ( I l L ) (|a> e x p ( - i j i K t r ) < l , \ ) | ^ i n t ( W c d ) | lc>exp(-i|?H t r) ® ( H t r ) < d \ ^ . (6.3.10) Making use of the appropriate i d e n t i t i e s f o r the transpose of the reference M i l l e r superoperator, ( H L ) A - l i 0 A . Q - 0 , ( 6 . 3 . U ) 200. ( X l * ) i l 0 - ® (H t r), (6.3.12) K t r ^ 0 = 1 1 ©"tr- (6.3.13) and ( i l L ) (AB) = ( ( f l L ) A ) ( ( i l L ) B ) , (6.3-14) o o o the symmetric kinetic cross section is then given by the expression, @ f S(ab|cd) = (if 2h 3/8vO^ I«>fi) (Htr)exp(-iptr)<b\\ ^ i n t ^ e d ) l I^ ^ ^ P t r ^ ^ t r ^ • which i s a p a r t i c u l a r matrix element of the frequency parameterized i n t e r n a l t r a n s i t i o n superoperator % i n t ^ ^ d ^ * According to Eq. ( 6 . 3 « 2),this superoperator i s the sum of two terms,and consequently the symmetric k i n e t i c cross section i s also the sum of two termsjnamely g) S(ab|cd) = <9 S» ( l )(ab|cd) + @ ' S , ( R ) ( a b | c d ) . (6.3.16) # as the scalar product The leading term i s f i r s t order i n 1* and can be written ( § S ' ( 1 ) ( a b | c d ) • dp 2h 3/8y.) ^ a > ® ( H t r ) •xp(-*^H t r)<b»|B o d^ . (6.3.17) 201. where the quantity l B c d ^ i s defined as Kd^ " ^ ( ^ © ( H ^ e x p C - i ^ H ^ K d l ^ . ( 6 . 3 . 1 8 ! The remaining term i s i n the form of a time c o r r e l a t i o n f u n c t i o n , @ S » ( R ) ( a b | c d ) = ( f V / S - r r / o j d s <?B&h I - 0 0 ' exp(-i(*V c d- ^ ) s ) | B c d ^ , ( 6 . 3 . 1 9 ) where the q u a n t i t y ^ ( ^ B a b J i s the product / B a b l = ^ > ® ( H t r ) e X p ( - * p H t r K b l l ^ - V ^ | a > ® ( H t r ) e x p ( - i p M t r ) < b l ( . ( 6 . 3 - 2 0 ) Two d i f f e r e n t approximation schemes are now consid-ered f o r the time c o r r e l a t i o n function. In one scheme, semiclassical-type approximations i n the s p i r i t of the DIA, SIA,DDIA and DSIA are presented. This i s done i n the remainder of the present chapter. On the other hand,the time i n t e g r a l i n Eq. (6 .3*19) can be performed i f a small posi t i v e parameter ^ i s introduced. This integration leads to a resolvent i n the generator*^, o ^dsexp( - i ( ^ c c j - ^ ) s ) -co 2 0 2 . o = lim ^lsexp(-i(W c d- £ + i ^ ) s ) = lim . i(^„ r t- i & ' + i ^ r 1 . ( 6 . 3 . 2 1 ) Using Eq. ( 6 . 3 . 2 1 ) , t h e symmetric k i n e t i c cross section becomes a matrix element of t h i s resolvent, (§' S» ( R )(ab|cd) = l i m Q + ( i p 2 h 3 / 8 l ^ . ) ^ B a b | ( < y c d - ^ + i ^ r l l B c d ^ ' ( 6 . 3 . 2 2 ) In chapter 7,the Zwanzig^ projection method w i l l be applied to t h i s resolvent and the relevant approximations: w i l l be considered and compared to the semiclassical-type approx-imations presented i n the remainder of t h i s chapter. 203 6.4.PHASE SPACE REPRESENTATION OF THE B.. OPERATORS The symmetrized k i n e t i c cross section has been written i n terms of the operators B... These operators a r i s e as state functions^Eq. 0 ( 6 . 3 . 1 8 ),and as observable functions,Eq. ( 6 . 3 * 2 0 ) . As a state function,the phase space representation of the operator B.. i s X J where .(r,p|r',p') i s the phase space representation 1KJL J 1 J — A / * * A * of the s u P e r ° P e r a " t o r * A s well as taking the phase space representation,the \ k ) ^ l | i n t e r n a l state representation of the operator B.. has been taken. The phase space represen-t a t i o n of the t r a n s l a t i o n a l state, S t r = ® ( H t r ) e x p ( - i p H t r ) , ( 6 . 4 . 2 ) has been defined as f»(r ,,p t). In analogy with the treatment i n chapters 4 and 5» t r a n s l a t i o n a l semiclassical-type approximations are defined i n the next section. In particular,the semiclassical-type t r a n s l a t i o n a l approximation involves taking the f u l l y quantal 1/-^ Q times the c l a s s i c a l Boltzmann weight, fp C L(r'»P*)• This c l a s s i c a l Boltzmann weight i n the phase space representation i s 204. ^ . C L ^ . P ' ) "/s'-rl ® ( H t r , C L > e x P ( - ^ H t r f C L ^ = h - 3 e x p ( - i p ' 2 / ^ -ipV 0(£'))® (*P' 2^ + V 0 ( r ' ) ) , (6.4.3) where the h~ 3 a r i s e s from the d e f i n i t i o n of the i d e a l phase space element ^^£'»P'| (2.2.29)* To simplify the re s u l t i n g expressions,the reference potential i s assumed to support no bound states so that the Heaviside function can be neglected i n the following. As with Eq. (2.3*23).the phase space representation of the t r a n s l a t i o n a l superoperator I K ± ; i j , y = 2TT h ' ^ r - r ' ) |dRexp(i( p'-p)-R/fe) < W S * * £ > S 3 1 - V^^r-iR) J k i ) . (6.4.4) The semiclassical approximation to Eq. (6.4.1) i s then B??IWr.P) = (2TT/h~ 7)fdp ,JdRexp(i(p ,-p)-R/n) -#/»V 0(r» (6.4.5) 205. Because the Heaviside function has been neglected,the i n t e g r a l over the momentum p* i s simply A/ Jdp'expUipp'2^ +ip'.R/&) = (kirp/p ) 3 / 2exp(-^R 2//3fc 2), (6.4.6) SCL so that the phase space representation of B. . as a state function becomes B ^ i j ( r . p ) = (2T/h 7)(4 1r /a/y8 ) 3 / 2 e x p ( - i ? V Q ( r ) ) j a R e x p ( - i p . R / n ) ( V l k i ( r H R ) ^ ( r - ^ ) S^) exp(-/*R 2/j3-n 2). (6.4.7) Eq. (6.4.7) can also be obtained from the semiclassical approximation to the phase space representation of the operator product V^expt-ija ) and i t s transpose. This i s accomplished by writing 2^ ^ as a commutator f i r s t and then taking the phase space representation;that i s , J k ' \ ^ ' H 6 X P ( "* f " t r , V 1 W | ki• <6•*•8> The phase space representation of the products of the 206. operators i n Eq. (6 .4 .8) can be written i n terms of the phase space representation of each operator,see,for example Leaf*®,or using Eq. (2.2.19)»the f i r s t product can be written as / r ' g l V l k i e x P ( - ^ H t r % =h-3jdRexp( - i p / R/ft) V l k i ( r+*5 > ^rtiR1exp(-I pH t r)Ir-ig). (6 .4 .9) whereas,the second product i s ^ P | e ^ - * P H t r ) V l j l ^ = h^JdRexpC-ig.^ V 1,.(r-^R). (6.4.10) l j l -One way of approximating these functions i s to consider semiclassical approximations to the position representation of the Boltzmann weight. For example,an operator F represen-t i n g the c l a s s i c a l Boltzmann weight can be obtained from the c l a s s i c a l phase space function h" Jexp(-ipp / ^ - 2 ^ V Q ( r ) ) . Indeed, i t i s just t h i s operator that w i l l reproduce Eq. (6 .4 .?) from Eqs. ( 6 . 4 . 8 ) , ( 6 . 4 . 9 ) and (6 .4 .10 ) . The operator F i s defined by using the phase space function h~ 3exp(-i|8 p 2 ^ - i ^ V Q ( r ) ) i n Eq. (2 .2 .2? )?that i s , F = h~ 3[drjd.p d ( r , p ) e x p ( - i p 2 ^ - * f ? V Q ( r ) 207. ) = h" 3(4T^V/9) 3 / / 2jdr jdRexp(-/*R 2/ph 2)exp(-i f v 0 ( r ) ) |r+*RVr-iRl. (6.4.11) In the position representation,this operator i s given by <r|Flr'> = h " 3 ( 4 u ^ / p ) 3 / 2 e x p ( - i p V 0 ( K r + r ' ) ) y a ( r - r ' ) 2 / ^ 2 ) , (6.4.12) a well known semiclassical f o r m u l a 1 1 . Replacing exp(-i jSH^) by i t s c l a s s i c a l counterpart F i n Eqs. (6 .4 .9) and (6.4.10) SCI reproduces Eq. ( 6 . 4 . 7)jthat is,the operator B. . i s formally J equal to B i f = ^ , Q ^ > F < J U (6.4.13) As well,the approximations of the next section CL also involve t r a n s l a t i o n a l l y c l a s s i c a l operators B. .. — J These operators are s t i l l considered quantally i n the i n t e r n a l states so that the h->>0 l i m i t i s applied to the quantal t r a n s l a t i o n a l motion only. To calculate the l i m i t , the p o s i t i o n R i n Eq. (6 .4 .7 ) i s defined as ifiz. This quantity i s then neglected i n the potential terms so that the r e s u l t i n g z i n t e g r a l i s simply the inverse Fourier 208. transform of Eq. ( 6 . 4 . 6 ) . The phase space representation of CL B. . as a state function i s then B°j(r,p) = 2irh _ l texp(-i|3p 2/> - * j 3 V 0 ( r ) ) The phase space representation of the operator B^. as an observable function i s also required. I t i s simply h 3 times the complex conjugate of the state function; namely .| I k > ( r , p ) ^ = h 3 ( B k i , i j ^ J » P ^ * « ( 6 . 4 . 1 5 ) The corresponding c l a s s i c a l and semiclassical-type approx-imations follow i n an analogous manner. The f a c t o r b? a r i s e s again from the difference i n the d e f i n i t i o n s of the i d e a l phase space elements |r,p^» and | r , p ^ . 209 6. 5.SEMICLASSICAL-TYPE APPROXIMATIONS AND THE DISTORTED WAVE BORN APPROXIMATION  6.5»a.DISTORTED WAVE BORN APPROXIMATION The exact quantal symmetrized k i n e t i c cross section i s given by the sum of two terms i n Eq. ( 6 . 3 . 1 6 ) . One of these terms tEq. ( 6 . 3 * 1 9 ).is i n the form of a time c o r r e l a t i o n function where the time dependence i s generated by the complete quantum L i o u v i l l e or von Neumann superoperator symmeterized k i n e t i c cross section involves the time i n t e g r a l of a matrix element . As i n chapter 5#the DWBA results when t h i s generator i s replaced by the reference generator «^ n» The DWBA o -oo o = "h~ 2JdsTr 4. ( £ ^ ^  S. V, exp(i(*V )s)V« S. J A t r db e t r lae A i' V J- x ec t r lec t r ( 6 . 5 . D where S^ . i s defined by Eq. ( 6 . 4 . 2 ) and where Tr.^ i s the tr a n s l a t i o n a l trace. To obtain t h i s expression,the reference 210. generator ' Q has been written as the sum of the i n t e r n a l free generator ^ ^ n i - and the t r a n s l a t i o n a l generator = *K^T + Using the c y c l i c property of the trace,the second term i n Eq. (6 .5»l) can be rewritten as o V 2 J d s T r t r S a c { V l e b S t r e x p ( i H t r S / M s t r V l d e 'OO exp( - i H ^ s/n) exp( i ^ d e s ) o - t - ^ d s l e x p ( i 4 d e s ) T r t r S^ S t / l d e e x p ( - i / t r s ) -OO leb t r CO = "^2Jds I exp(i^ eds) ^ VledStrlexp(i/trs) o V l e b S t r ^ f • (6.5.2) where i n the l a s t l i n e the time s has been changed to -s. In a s i m i l a r manner,the fourth term becomes « -*- 2/d STr t r. exp( i « a o s ) V l d b S t r e x p ( i / t r s ) V l a e S t r - OO CO = ^ " 2 / d s T r t r e x p ( i ^ s ) V ^ S ^ e x p t i f^s ) \ ^ \ r -( 6 . 5 . 3 ) Now using the i d e n t i t y , e x p ( i o t t r s ) V l d b S t r -(exp( i f ^ a ) V l d b ) ( e x p ( • ^ g ) s ^ } 211. = ( e x p ( i / t r s ) V l d b ) S t r , ( 6 . 5 . 4 ) and the c y c l i c nature of the t r a c e ,Eq. (6 .5*3) can be r e w r i t t e n as -1i"*2J dsexp( iCJ s)Tr. V„ , , S . e x p ( i * . s )V„ S. -oo a c t r l d b "fcr t r l a c t r OO - . h ^ / d s e x p U ^ s ) / v i c a S t r | e x p ( i < / t r s ) ( s t r V l d ^ ( 6 . 5 - 5 ) Thus the time c o r r e l a t i o n p a r t of the DWBA c ross s e c t i o n can be w r i t t e n as o ( 9 S ' ( R K D W B A ( a b \ c d ) - (fW/2/M | ( S d b ] d 8 - 0 0 «P< ^ e c 3 > J > i e a S t r M 1 ^ t r s > 1 v l . o s t r ^ f + S a c ( d s e x P ( i H ^ ] -( i j5 2hTr (Jdsexp( i O ^ s ) +Jdsexp( i ^ s ) ) -oo X v l = a S t r l e x P ( i A r s » ) S t r V l d b | • < 6-5- 6 ) For population effects,a=b and c=d,the integrals add and the DWBA symmet rized., k i n e t i c cross section becomes oo ^S,DWBA ( a a j c c ) m ify*hv^)^at%M£eXj>UQetia) - o o 212. / Vl.. Strl-^i'tr»> l vi« str^ - e x p f i ^ s ) / v i c a S t r l e x p ( i ^ r S ) | S t r V l o a ^ ). ( 6 . 5 . 7 ) 213. 6.5.b.SEMICLASSICAL-TYPE APPROXIMATIONS As i n chapter 5.the time c o r r e l a t i o n function.Eq. (6.3*19),is i n a form appropriate f o r making se m i c l a s s i c a l -type approximations. Here,because i n t e r n a l states are present, only distorted wave Born semiclassical-type approximations are considered,since t h i s assumption greatly s i m p l i f i e s the dynamics. As well,to simplify the expressions,the reference p o t e n t i a l i s assumed to support no bound states so that the Heaviside function can be neglected i n S^ r and,as well,only population e f f e c t s are considered. The f u l l quantal nature of the i n t e r n a l states i s retained. Differences i n the t r a n s l a t i o n a l motion between quantal and c l a s s i c a l mechanics a r i s e i n the dynamic-like interference effects due to exp(i 0^-j.s) .and i n the s t a t i c - l i k e interferences i n the B..'s. This s i t u a t i o n leads n a t u r a l l y to two d i f f e r e n t semiclassical-type approximations,the k i n e t i c distorted wave dynamic interference approximation (KDWDIA) and the k i n e t i c distorted wave s t a t i c interference approximation (KDWSIA). In the KDWDIA,the s t a t i c interference e f f e c t s are ignored while the dynamical interference effects i n the evolution generator Q are retained. In ignoring the s t a t i c i n t e r f e r e n c e s , a l l the t r a n s l a t i o n a l e f f e c t s i n the CL B^..'s are taken as c l a s s i c a l ; t h a t is,B^^,Eq. (6.4.14),is used instead of the quantal B9.. In the KDWDIA,the cross section,Eq. (6.5»7).becomes oo ^S,KDWDIA ( a a | c c ) = ( i | l 2 h T ^ ) J d s ( £ S a c e x p ( i * e a s ) 214. a<Vlea, C L e x P ( ' ^ H t r , C L * I e xP< i ^ t r , Q ) Kea,CL 5 X P ( - ¥ H t r , C L % " expd^ s) ^ ^ e x p ^ H ^ ^ ) ! 2 xP ( i^tr,Q s )l e xP (-¥ Htr,CL ) V l c a,CL^ >• ( 6 - 5 . 8 ) Further evaluation of t h i s approximation requires knowledge of the exponential of the generator *^- R Q . . This approx-imation i s not pursued here. In the KDWSIA,the dynamic interference effects are ignored while the s t a t i c interference effects i n the B ^ j ' s are retained. Here,the quantal generator e£. n i s replaced, ~cr, w by the c l a s s i c a l generator * ^ j . r and the KDWSIA cross section becomes oo ^ S . K D W S I A ( a a j c c ) = ( i p 2 h i r ^ ) J d s ( |S a cexp(iW e as) ^ l e a . Q ^ - ^ t r . Q ^ ^ ^ t r . C L ^ K l e a . Q e x p ( - i p H t r > Q ) ^ - e x p ( i ^ c a s ) ^ < V l c a t Q e x p ( - f p H t r t Q ) \ ^ ^ t r . C L ^ l ^ - ^ t r . Q ^ l c a . Q ^ >' ( 6 ' 5 * 9 ) Evaluation of t h i s approximation requires knowledge of the reference quantal Boltzmann f a c t o r expC-ijJH^ Q ) . A further approximation to the K D W S I A can be made by replacing the f u l l y quantal B 9 . by the semiclassical B?9 L,Eq. (6.4.7). In 2 1 5 . t h i s case,the superoperator V: „ i s retained while the reference Boltzmann f a c t o r i s taken as c l a s s i c a l . As shown i n Eq. ( 6 . 4 . 1 3 ) , t h i s approximation i s equal to replacing Z ^ Q e x p ( - | p H t r ^ Q ) by ^ Q F where F i s given by Eq. ( 6 . 4 . 1 2 ) , This approximation i s termed the k i n e t i c distorted wave c l a s s i c a l t r a j e c t o r y approximation (KDWCTA) and i t s cross section i s defined as (gS.DWCTA ( a a l c c ) = ( | p 2 h T / ^ ) J d s ( | S a c e x p ( i U e a s ) - O B ^ V l e a . Q P | e x p ( i ^ r j G L s ) | v l e a Q F ^ - e X p ( W o a s ) ^ V l c a y l e * P ( i ^ t r . C L s M F V l c a > Q % ) . ( 6 . 5 - 1 0 ) The KDWCTA emphasizes the t r a n s l a t i o n a l quantal effects inherent i n n while suppressing a l l other quantal l ,y e f f e c t s . In Sec. 6 . 6 the KDWCTA w i l l be written i n terms of the reference c l a s s i c a l t r a j e c t o r i e s . As well,a straight l i n e approximation to the motion w i l l be considered. When the l a s t vestiges of the quantal e f f e c t s i n SCL the t r a n s l a t i o n a l motion are removed;that i s , r e p l a c i n g B.. J by BT\ i n Eq. ( 6 . 5 . 1 0 ) , o r f o r that matter,replacing the quantal generator ot^.r Q by the c l a s s i c a l generator Q J , i n Eq. ( 6 . 5 » 8),a " f u l l c l a s s i c a l " approximation r e s u l t s . Of course,the i n t e r n a l states are s t i l l considered quantally. In t h i s approximation,the k i n e t i c distorted wave f u l l c l a s s i c a l approximation (KDWFCA),the cross section i s 216. o o ( g S t K D W F C A ( a a j c c ) m ( i j l 2 h T r ^ ( ) | d s ( £ S a cexp(iU) e as) - o o ^ V l e a , C L e x P ( " * P H t r , C L > l e xP ( 1 .CL S ) IVlea,CL e x P < - ^ H t r , C L % " e x P ( W e a s ) < ^ V l c a , C L e x P ( - * ? H t r , C L , l e x P ( i ^ t r , C L s ) i e x P ( - ^ H t r . C L ) V l c a , C L ^ >' <*-5.U> As with the KDWGTA,in Sec. 6.6,the KDWFCA w i l l be written i n terms of the c l a s s i c a l reference t r a j e c t o r i e s . The straight l i n e approximation to the KDWFCA i s the CAA of; Oppenheim and Bloom-'. 2 1 7 6.6.KDWFCA,KDWCTA AND STRAIGHT LINE APPROXIMATIONS  6.6.a.KDWFCA AND KDWCTA The KDWFCA cross section i s given by Eq. ( 6 . 5 . 1 1 ) 1 . 1 In the phase space representation i t becomes ^S,KDWPCA ( a a | c o ) . ( i f2 T^2,ji B C l y | d p ( £ — (to e x p ( i W e a s ) e x p ( . * p p 2 ^ - & V 0 ( r ) ) V l a e ( r ) V l e a ( r b ( s ) ) - e x p ( i ^ a s ) e x p ( - i p 2 ^ - | 3 V 0 ( r ) ) V l c a ( r ) V l a c ( r 0 ( s ) ) , ( 6 . 6 . 1 ) where the po s i t i o n r n ( s ) , a l o n g with the momentum p n ( s ) , s a t i s f y Hamiltons equations of motion f o r the reference system subject to the i n i t i a l conditions r Q ( 0 ) = r and p n ( 0 ) = p. To obtain t h i s expression,*Eq. ( 6 . 5 . 4 ) f o r the c l a s s i c a l reference system has been used;that i s , h 3 / r ' 5 1 e x p ( 1 ' t r , C L S } \ e x P ( "* f H t r , C L ^ = exp(-ijS p 0 ( s ) 2 / ^ - * p v 0 ( r 0 ( s ) ) ) = e x p ( - i p p 2 ^ - i ^ V Q ( r ) ) . ( 6 . 6 . 2 ) The KDWCTA cross section,Eq. ( 6 . 5 . 1 0 ) , i n v o l v e s the phase space representations of the operator products V^-j Q F and FV i.. n which are obtained by using Eq. ( 6 . 4 . 1 2 ) i n 113,y Eqs. ( 6 . 4 . 9 ) and ( 6 . 4 . 1 0 ) respectively. Using these equations, 218. the KDWCTA cross section becomes ^ s , K D w c T A ( a a l c c ) „ ( a S ^ a ^ e ^ ^ l ^ P j j ^ j ^ . exp(-A(RZ+R'Z)/fiii2 - ^ V Q ( r ) +ig.(R-S')/*)( f $aG  V l a e ( £ + ^ ^ e a ( £ o ( s ) - ^ ' )exp(i« as).exp(i«caa) v l a c ( £ + * 5 ) v i C a ^ 0 ( 8 ) ^ 5 , » . ( 6 . 6 . 3 ) where,again,Eq. (6.6.2) has been used. In the c l a s s i c a l l i m i t of the t r a n s l a t i o n a l motion,the KDWCTA becomes the KDWFCA. This r e s u l t follows when R and R' are written as nz and nz' and t h e i r e f f e c t s on the potential terms are neglected. Performing the z and z' int e g r a l s leads to the KDWFCA cross section,Eq. ( 6 . 6.1). 219 6.6.b.STRAIGHT LINE APPROXIMATIONS The KDWFCA and KDWCTA cross sections have been written i n terms of the reference t r a j e c t o r i e s r Q ( s ) and P Q ( S ) . Further approximations to the c l a s s i c a l dynamics are possible. A p a r t i c u l a r l y simple approximation i s to assume straight l i n e motion where the t r a j e c t o r i e s become ,SL and rQia)01J = r + ps/^ , ( 6 . 6 . 4 ) P 0 ( s ) S L = p. ( 6 . 6 . 5 ) Because the reference motion has been approximated,Eq. ( 6 . 6 . 2 ) can no longer be used. The straight l i n e f u l l c l a s s i c a l cross section,KSLFCA,becomes ( § S , K S L F C A ( a a l c c ) = (ip2-rr^h2)Jdsjdr[dp e x p C - i / i p ^ . - i / l V ^ r j - i p V ^ r + g s ^ J X { & a c e x p ( i 6 ; e a s ) V l a e ( r ) V l e a ( r + p s ^ ) - e x p U ^ s ) ^ a c ^ ^ i c a ^ ^ ^ ' ( 6 - 6 ' 6 ) a r e s u l t which Snider and Turner' have shown to be equi-valent to the CAA defined by Oppenheim and Bloom-*. The KSLFCA i s also equal to replacing the generator <£- r^ Q by the free generator i n the KDWDIA cross section,Eq. 220. ( 6 . 5 » 8 ) . The KSLCTA cross section represents a p a r t i c u l a r semiclassical extension to the CAA. I t i s ( g S , K S L C T A ( a a l c c ) = ( 2 5 ^ V 2 / ? h 8 ) ^ s | d j j d p | d g J d ^ , exp(-/*(R 2+R , 2)/pfc 2 - i p v 0 ( r ) -ipV 0(r+ps^c) +ip.(R-R')/fe)( | S a c e x p ( W e a s ) Y l a e ( r + i R ) V l e a ( r + p s ^ -*3' ) - e x ^ i ^ a s ) V l a c ( r * R > V l c a ( r P s ^ + * S , ))• ( 6 . 6 . 7 ) 221. 6.7.DISCUSSION Kinet i c cross sections have been written i n terms of the t r a n s i t i o n superoperator, 7^, which, i t s e l f , has been written i n terms of the double potential formalism of chapter 5. In the present chapter,semiclassical-type approximations to the r e s u l t i n g time c o r r e l a t i o n functions have been presented. In particular,dynamic interference ef f e c t s were emphasized i n the KDWDIA while s t a t i c i n t e r -ference e f f e c t s were emphasized i n the KDWSIA. A further approximation was made to the KDWSIA where the Boltzmann weights were taken as c l a s s i c a l . This was termed the KDWCTA. Rounding out th i s approximation scheme,which i s summarized i n Table 6.1,the KDWFCA was defined where a l l the trans-l a t i o n a l e f f e c t s were taken as c l a s s i c a l . In a l l t h i s scheme,the i n t e r n a l states are treated quantally. The KDWCTA and KDWFCA were presented i n terms of the c l a s s i c a l reference t r a j e c t o r i e s . Taking t h i s motion as a straight l i n e resulted i n the KSLFCA and the KSLCTA. The KSLFCA cross section i s equal to the CAA of Oppenheim and Bloom-* while the KSLCTA represents a semiclassical extension to the CAA. In the next chapter,rather than considering the time dependence e x p l i c i t l y , t h e k i n e t i c cross section i s written i n terms of the resolvent f o r the generator £ . 8 Zwanzig projection methods are applied to t h i s resolvent and the r e s u l t i n g expressions are compared to the semi-c l a s s i c a l - t y p e scheme presented i n th i s chapter. K i n e t i c Cross Section (QM) KDWBA(QM) KDWDIA SLA I KDWSIA SLA KSLSIA SLA KSLCTA KDWCTA KDWFCA if SLA KSLFCA = CAA Table 6.1 QM = Quantum Mechanics SLA = Straight Line Approximation CHAPTER 7 KINETIC CROSS SECTIONS:PROJECTION TECHNIQUES 224 7.1.INTRODUCTION So f a r , t h i s thesis has emphasized the time depen-dent nature of c o l l i s i o n processes. In particular,the d i f f e r e n t i a l cross section f o r both c l a s s i c a l and quantum mechanics was defined from a time dependent viewpoint i n chapter 3.while i n chapters 4 and 5 dynamic and s t a t i c semiclassical-type approximations have been presented. Kin e t i c cross sections were defined i n chapter 6 and,again, s t a t i c and dynamic semiclassical-type approximations were formulated. Alternate to emphasizing the time dependence, these cross sections can also be written as matrix elements of resolvents,see,for example,Eq. (6.3*21). In t h i s form, 1 2 Zwanzig projection techniques can be applied. Levine has used these projection techniques on the quantal wave functions involved i n the d i f f e r e n t i a l cross section. Having the present formulation allows these projection techniques to be applied d i r e c t l y to the d i f f e r e n t i a l cross section as written i n Eq. (4.2.6). Again,from t h i s viewpoint,the technique i s applicable to both c l a s s i c a l and quantum mechanics. As well,these projection techniques can be applied to the k i n e t i c cross sections of chapter 6. I t i s the l a t t e r which i s considered here. The k i n e t i c cross section,for i n t e r n a l state trans-i t i o n s , has been written i n terms of the resolvent assoc-iated with the complete generator £ i n Eq. (6.3*21). In Sec. 7*2,the projection technique i s applied to t h i s resolvent. The r e s u l t i n g formula f o r the cross section i s 225 exact f o r whatever resolvent used,a f a c t which i s demon-strated by showing how the projected r e s u l t can produce the o r i g i n a l resolvent. Since the projection technique i s exact f o r any evolution generator,the semiclassical-type approximations of chapter 6 can be expressed i n projection formula terms,see Sec. 7«3» In Sec. 7«4,approximation schemes d i f f e r e n t from the semiclassical-type approximations of chapter 6 are presented. I t i s shown that the r e s t r i c t i o n to the compli-mentary space i s important when evaluating memory kernels and that neglecting t h i s r e s t r i c t i o n can lead to poor r e s u l t s . Useful approximations then involve the component i n the complimentary space of the generator of the motion being considered. In Sec. ? .5»it i s shown how a 1-dimen-sio n a l projection leads to a generator ( i n the complimentary •a space) with a generalized separable potential . 226. 7.2.GENERAL PROJECTION FORMULAE In chapter 6.the time dependent nature of the symmeterized k i n e t i c cross section,Eq, (6.3.16),was empha-sized. There,the symmeterized k i n e t i c cross section was written as the sum of two terms,one being f i s t order i n 2^,Eq. (6 .3«17)$the other being a time c o r r e l a t i o n function,Eq. (6.3.19). The time dependence of t h i s c o r r e l a t i o n function was determined by the generator ^ of the complete motion. A l t e r n a t e l y , t h i s time c o r r e l a t i o n function can be written as a matrix element of the resolvent associated with the generator <^T,Eq. (6.3*21). The contribution to the ki n e t i c cross section i s then Eq. (6.3.22), ( g S » ( R ) ( a b | c d ) = lim + < i f 2 h 3 b a b b G d / 8 T A ) where the unit operators B., are defined by (7.2.2) involving the normalization constants b. . given by (7.2.3) Rather than approximating the motion defined by the exponential of the generator as was done i n chapter 6, 227. approximations to the resolvent can be considered. One method of formulating approximation schemes for the resol-vent i s to use the Zwanzig1 projection technique. In this method,a projection superoperator & i s defined such that i t projects onto the particular set of B..'s of interest; that i s , <?! . .=! . . , (? 2 = <? , (P+= <P . (7 .2 .4 ) The matrix element of the resolvent i n Eq. (7.2.1), (7 .2 .5) then involves only the projected resolvent 0 (Wcd- ^H^)"* 1^ 0 By algebraic rearrangement,the projected resolvent can be written 1 in terms of the motion within the ® space, generated by (P£® ,and a memory due to the motion gener-ated in the orthogonal or complementary space by (l-(?) <^(1-<P) jthat i s , = (0ci-<?J<? +i^fW o d)Hjr 1(?. (7 .2 .6 ) where the memory kernel i s 228. 2^(# c d> = i t f * ^ ( l - ( ? ) ( u ) c d - ( l - ( P ) d f ( l - P j + i ^ ) " 1 ( 7 . 2 . ? ) Two separate cases a r i s e . I f the observable operator and the state operator are equal,then the projection defined by Eq. ( 7 . 2 . 4 ) can be one dimensional. This case a r i s e s when a=c and b=d. On the other hand,if the observable and state operators are not equal,then the projection must be at least two dimensional. This case arises f o r a l l other i n t e r n a l state combinations. Of course,the projection,Eq. ( 7 * 2 . 4 ) , i s not unique,as,for example,it could be the projection onto next two subsections,the one and two dimensional cases are considered. the f u l l set of operators f o r a l l i and j . In the 229. 7.2.a.THE ONE DIMENSIONAL CASE When a=c and b=d,the simplest choice of a projection superoperator i s the one dimensional case,where the pro-ject i o n i s e x p l i c i t l y In t h i s case,the matrix element of the resolvent,Eq. (7.2.5)»becomes " ( Wab "^ablab • i\.,.b«'.t> ,* 1?>' 1' <7-2-9> where the general matrix e l e m e n t ^ i * - ^ i s defined by and where the matrix element of the memory kernel i s Mab.ab^ab> ' i ) ( « a b - ( l . ( P )^(l-<? ) +i^)~ 1(l-<P ) £ l B a b ^ • ( 7 . 2 . 1 1 ) The contribution to the cross section i s then S » ( R ) ( a b l c d ) = lim . ( i * 2 h 3 b . 2/8ir/x) ^ 0 ' a D ' 230. Cab " *ab!ab + i Mab.ab ( 4V W1- <'"-2-12> 7.2.b.THE TWO DIMENSIONAL CASE For the general two dimensional casejthe projection superoperator,Eq. ( 7 . 2 . 4 ) , i s -bod.abl5od\ ^ 5ab\- bab,cdl5at^  fcl >.<7.2.13> where the scalar product b a b c d i s bab.od " = '"cd.ab'*- ( 7 . 2 . 1 4 ) The k i n e t i c symmetrized cross section involves the matrix element, r ' W d • A b | ( ^ c d ) r l l 5 c d V ^.2.15) where the superoperator °& (^ e (j) i s defined as < ^ ( ( y c d ) = ^ c d " +i7/?(Vcd) +i<y. ( 7 . 2 . 1 6 ) , In terms of the matrix elements of «Cr (^ c d)»f or example, Dab.od - ^ r t l ^ c d ' ^ o ^ l f the inverse matrix ( D _ 1 ) i s -D ab.ab cd.cd ab.cd cd.ab ) -1 231. " ^ c d ^ a b . c d -"ab.cd + i " W d ( 6 , c d ) - ( ? ' 2 ' 1 7 ) Dcd,cd ~ Dab,cd \ -D cd.ab Dab.ab/ ( 7 . 2 . 1 8 ) Thus,the contribution to the k i n e t i c symmetric cross section due to the resolvent i s ( g S ' ( R ) ( a b | c d ) = - l i ^ ( i p 2 n \ b b c d / 8 i r ^ ) D a b , c d ( Dab,ab Dcd,cd- Dab,cd Dcd,ab ) ( 7 . 2 . 1 9 ) 232. 7.2.e.EQUIVALENCE OF THE MEMORY EXPRESSION TO THE ORIGINAL  PROJECTED RESOLVENT Eqs. (7.2.12) and (7.2.19) f o r the symmetric k i n e t i c cross section are exact since Eq. (7 .2.6) f o r the projected resolvent i s an exact expression. I t i s exact f o r whatever projection i s used and f o r whatever generator i s used,as long as CP contains the appropriate "^^y This f a c t i s used i n the following section to obtain the projection forms of the semiclassical-type approximations of chapter 6. That Eq. (7 .2 .6) i s exact i s now demonstrated further by s t a r t i n g with the right hand side of Eq. (7 .2.6) and then reproducing the l e f t hand side. The product of the resolvent f o r the ( l - ( ? ) space generator,(l-(?) of (1-<P ),and ( l - ( P ) o f ^ i s assumed to be a power series i n the couplings (1- ) «£n(? ; that i s , (4> c d-< 1- CP ) £{ 1- <? 1- <?) < O O = £ ( l - ( ? )^ n0>A , (7.2.20) n=l n where the c o e f f i c i e n t s A n are "matrices" i n the CP space. I f , f o r example,the projection i s l-dimensional,then the c o e f f i c i e n t s A^ are simply numbers whereas i f the pro-je c t i o n i s 2-dimensional then the A^'s are 2-dimensional c o e f f i c i e n t matrices. To obtain expressions f o r the coe-f f i c i e n t s A n,Eq. (7.2.20) i s multiplied through by the inverse of the resolvent and the expansion c o e f f i c i e n t s 233. are equated on the basis that the coupling terms {l-Q ) £n(? are a l l independent;that i s . n=l cd "? ) An -(!-<? ) ^ n + 1 ( ? A n + (1-CP)SC(? C ? ^ n ( p A n ) . (7.2.21) For n ^ 2 , t h i s r e s u l t s i n the recurrence r e l a t i o n A n = ( < W c d + i < ? r l A n - l » n ^ 2 » (7.2.22) whose solut i o n i s A n = ^ e d * 1 ^ 1 " " * ! ' (7.2.23) For n=l,using Eq. (7 .2 .22),the c o e f f i c i e n t of ( l - ( P ) ^ C ? gives P " ( £ J o d + i ^ ) 0 > A l + ^ # n=l n oo = ((? +(?£ (^/(W c d+i^ ) N ( P X ^ + i y j A j ' = ( C ? + G Y i -i)(? X ^ I ^ A , • ( W c d + i 7 ) 2 ^ ( ^ c d - ^ + i t r l ^ A l - (7.2.24) Rearranging Eq. (7.2.24),the c o e f f i c i e n t matrix A^ becomes 234 On using Eqs. ( 7 . 2 . 2 0 , 2 2 and 2 5 ) ,the memory kernel,Eq. ( 7 * 2 . 7 ) , i s calculated ( a f t e r a few algebraic steps) to be ( 7 . 2 . 2 6 ) The r i g h t hand side of Eq. ( 7 . 2 . 6 ) involves the inverse of the superoperator *^ (^ c d),Eq. ( 7 . 2 . 1 6 ) ,which,using Eq. ( 7 . 2 . 2 5 ) , i s ^ c d > " " c d - cd> +(^(^ c d- A i ^ r 1 ^ ) ' 1 = ( ^ ( ^ c d - / + i | ) - 1 ^ ) - 1 . ( 7 . 2 . 2 7 ) Inverting Eq. ( 7 . 2 . 2 7 ) gives the l e f t hand side of Eq. ( 7 « 2 . 6 ) ,a re s u l t which v e r i f i e s the exactness of Eq. ( 7 . 2 . 6 ) . 235. 7.3.RELATION TO THE SEMICLASSICAL-TYPE APPROXIMATIONS The exact k i n e t i c cross section was written i n Sec. 7.2 i n terms of a memory superoperator using Zwanzig's 1 projection technique. This method i s exact f o r whatever form of the B..'s that are used and f o r whichever p a r t i c -u l a r generator i s being considered. In particular,the distorted wave Born approximation (see Sec. 6 .6) r e s u l t s when the complete generator i s replaced by the reference generator In t h i s case,the resolvent can be written i n terms of the t r a n s l a t i o n a l reference generator «^- r as / I a b l ^ c d ^ 0 + i ? ) " 1 l B c d ^ - i ^ f . a b l ^ e d ^ e f - < r + ^ r M 5 e f . c d ^ r ' ( 7 . 3 . D Taking the operators B. . as quantal,l9. ,as well as 1j » KX X j,KX taking the quantal generator, Q,gives the f u l l quantal distorted wave Born approximation given i n Sec. 6.5«a. A t r a n s l a t i o n a l projection superoperator, (?®,is defined which projects onto the operators l9 . X J , KX # Q l i j , k l - ^ . k l ^ ^ ^ ^ ^ ^ ^ ^ ' ( 7 . 3 . 2 ) This projection superoperator can be i n f i n i t e dimensional i f i t projects onto the set of operators £l9. ./\ f o r a l l i , j and f o r kl=ab and kl=cd,or i t can be 1- or 2-dimensional i f each ef combination i s treated separately. The matrix 2 3 6 . element of the resolvent,Eq. (7.3.1).becomes where the memory kernel,Eq. ( 7 » 2 . 7 ) » i s ( 7 . 3 . 4 ) The exact DWBA k i n e t i c cross section can then be written as rz* S,(R),DWBA, , 1 n . , • * 2 . 3,Q VQ / a ^ t x (©> (ab/cd) = l i m + Up h vb a bb^ d/8-i9^) 0 e f ( o V ef,ab« x cd ef ^ t r . Q A * ^ T R Q cd <sf' + i 7 r l ' ^ f , c d ^ ' ( 7 - 3 . 5 ) plus the f u l l quantal f i r s t order term.Eq. ( 6 . 3 . 1 7 ) . The KDWDIA was defined i n Sec. 6.5.b by replacing the quantal operator l9 . with i t s t r a n s l a t i o n a l c l a s s i c a l 1 j • K J -237. QT counterpart B ^ fel while re t a i n i n g the f u l l quantal trans-l a t i o n a l generator Q» In the projection formulation, t h i s corresponds to replacing the quantal projection by the t r a n s l a t i o n a l l y c l a s s i c a l projection ( P C L f o r the QT functions B.. For population effects,the KDWDIA cross 1 J , KJ. section i s then /^S,KDWDIA/ I \ . . / . />21_3n_GL, C L / o _ x (0 ' (aalcc) = lim . (lfi h"V b /8TTM) 0 aa cc an expression that i s equal to Eq. ( 6 . 5 . 8 ) . On the other hand,the KDWSIA was defined by replacing the t r a n s l a t i o n a l quantal reference generator ^^T Q with i t s c l a s s i c a l counterpart, *t^- r Q-^  while retaining the quantal l 9 . . The KDWSIA cross section i s then @S.BWSIA ( | | a l o e ) . . l i B + 1 / 8 ^ , + i V " 1 | 5 ^ f . o c ^ • ( 7 - 3 - 7 ) f o r population e f f e c t s . This i s equal to Eq. ( 6 . 5 * 9 ) . On replacing B ^ f k l by B ^ k l , t h e KDWCTA, 238. /^S,KDWCTA/ , v ... , . 2, 3^SCL,.SCL/o % (G) (aajcc) * lim ( 1 6 h J b ^ /8TT/A.) ^ ^ Q i aa cc ' i s obtained. This hierarchy of semiclassical-type approx-imations i s comple obtain the KDWFCA. —S CL pT ted by replacing B7 . , , with B. . - to 1 J » K l 1 J | iCl © S,KDWFCA/ I \ _ . /-*2. 3J3L. CL/o,. * (aa|cc) = lim , (lA'hW„/8TT>L) ^ 0 a a c c ' - < ? ° V + r T ^ C L  e f t f * x e f . a a T fe <*tr,CL + i ^ ^ r C L ( W f e ) + i ? , " 1 ) 5 e f , o o ^ • ( 7 . 3 . 9 ) In Sec. 6.6.b,straight l i n e t r a j e c t o r y approximations were presented. These approximations involve replacement of the reference t r a n s l a t i o n a l generator ^ v with the free t r a n s l a t i o n a l generator • The straight l i n e approximation to the quantal DWBA, @ S ' S L ( a a | c c ) = l ^ o + C i p 2 h 3 b ^ c / 8 ^ ) + i t r l ^ f . o o % • (7-3.10) 239. uses the memory kernel U - 0 " 3 ) ^ . ^ . ( 7 . 3 . U ) In p a r t i c u l a r , r e p l a c i n g the t r a n s l a t i o n a l quantal reference generator * ^ T R Q with Q i n the KDWDIA or,for that matter,replacing the c l a s s i c a l t r a n s l a t i o n a l reference generator c?^r with Q^,defines the straight l i n e f u l l c l a s s i c a l approximation, @ S . K S I « A ( a a l o c ) . u . + U a V b ^ / S ^ ) q-*0 ' aa cc / This expression i s equal to Eq. ( 6 . 6 .6 ) and thus i s the projection formula equivalent to the GAA of Oppenheim k and Bloom . In the next section,approximations to the exact resolvent express!on,Eq. (7»2«6),are presented which d i f f e r from the semiclassical-type approximations of chapter 6. 240. 7.4.RESOLVENT APPROXIMATIONS 7.4.a.IMPORTANCE OF THE (1-(P) SPACE MOTION The expression f o r the projection of the complete resolvent (^ c d- d+iy)"1 ,Eq. ( 7 . 2 . 6 ) . i s known to be exact .a r e s u l t which has also been demon-strated i n Sec. 7.2.e. The re l a t i o n s h i p between t h i s equality and the semiclassical-type approximations of chapter 6 has been given i n Sec. 7 » 3 . Other approximations to the memory,Eq. ( 7 . 2 . 7 ) . a r e possible. An obvious approx-imation i s to neglect the projectors i n the memory kernel;that is,the exact memory kernel i s replaced by the memory kernel ( 7 . 4 . 1 ) ^ N ^ c d ) " i ^ ^ c d - ^ + i t ) " 1 ^ - ( 7 . 4 . 2 ) This approximation i s poor as w i l l now be shown. The resolvent i n Eq. ( 7 . 4 . 2 ) i s mul t i p l i e d fore and a f t by the superoperator and,thus can be rewritten as ^(4/cd-/+i^rV 241. « < * c d - ^ + i V - 2 ( " c d + ^ (7.4 .3) The memory kernel,Eq. ( 7 .4.2),is then - - i ( 4 / c d + i ^ - i ^ 5 +i(^cd+i<?)2^(^cd-/+it)"1^P. (7.4.4) With Eq. (7.4.4) as an approximation to the exact memory ^ ( ^ c d ) , t h e complete resolvent i s approximated by ^ c d - ^ + i ? r 1 ^ ~ U / e d + i f ^ i ^ V o d ) ) " 1 ^ ( ^ c d + i ? r l ( 2 - ( ^ c d + i ' ? ) ^ ( " c d - ^ + i j ) " 1 ^ ( 7 . 4 . 5 ) In the l i m i t ^ •* 0 + , the cross section i s then proportional to the inverse of the frequency. Furthermore.if the frequency i s zero.then the cross section w i l l be i n f i n i t e i n t h i s approximation. Thus,neglecting the ( l - C ? ) projectors appears to be a poor approximation. Since retention of the (1-0) motion seems important,then other approximations of the same form as Eq. (7.4.2),for example 242. fy{Vcd)~ i ^ c d 4 + i ? ) - W , ( 7 . 4 . 6 ) do not seem appropriate. 7.4.b.RESOLVENT APPROXIMATIONS As demonstrated i n Sec. 7*4.a,the space nature of the memory kernel i s important. For example, the DWBA memory kernel i s given by Eq. ( 7 . 3 » 3 ) . Rather than neglecting the (1-<P^) nature of t h i s memory, the resolvent i n th i s expression can be replaced with a projected free resolvent»that i s , " i ^ V t r , Q ( i - ( ? 9 ) ( ^ e - ( i - ^ 9 ) ^ t r (l-0>«)H-j)-1(l-<?Q)<^triQ(PQ, (7A.7) f o r population changes. This approximation represents an extension of the straight l i n e memory,Eq. (7»3«10),and i s termed the free memory resolvent (FMR) approximation. This approximation can be applied to the KDWDIA, KDWSIA,KDWCTA and the KDWFCA. In particular,the KFMRFCA cross section i s 243. ^f .eo% • (7 . 4.8) This represents an extension to the KSLFCA (CAA). Sim i l a r approximations to the exact memory,Eq. (7»2.7),can be made by s e l e c t i v e l y approximating the d i f f e r e n t generators involved. These approximations are not pursued here. This chapter i s concluded with some comments on the eigenfunctions of the (l-(P) space free generator since they can be used to evaluate these approx-imations. 244. 7. 5. COMMENTS ON THE EIGENF UNCTIONS OF (1-(? ) fy^ ( 1 - 0 )  7.5.a.MgfcLER SUPEROPERATOR In Sec. 7.4 the importance of the (1-0*) projectors i n the memory kernel was demonstrated. The se m i c l a s s i c a l -type approximations of chapter 6,when written i n the projection formulation and the FMR approximation,involve the resolvents of (1-(P ) £ . (l-(? ) and ( l - P j t y l (l-CP). To X T X T evaluate these resolvents,the eigenfunctions of these superoperators can be used. Here1, the eigenfunctions of are considered. For s i m p l i c i t y the projec-t i o n i s taken as one dimensionalithat i s , Q - 1 5 e f .aa>^ / 5 e f .aa.1 • (7.5.1) where B f a a can be the quantal operator a a . t h e semi-* —SCL ' PI c l a s s i c a l operator B „ or the c l a s s i c a l quantity . ex,aa ex,aa The eigenfunction equation i s ( l - ^ ^ d - ^ f e ) ^ - v*l u<y.*%- (7-5.2) I t i s convenient to introduce.a L i o u v i l l e or von Neumann superoperator * C = ^ t r + ^P» ( 7 - 5 ' 3 ) where the pot e n t i a l superoperator i s defined by 245 ^ - - ' K - K * ^ . 5 . 4 ) The superoperator e^ p i s formally s e l f - a d j o i n t while the potent i a l superoperator approaches zero f o r large positions. Thus a M i l l e r superoperator f o r can be considered;that,is S i = iim e x p ( i ^ l t ) e x p ( - i / ? l t ) . ( 7 - 5 . 5 ) Eigenf unctions U(v ,K),of ^ p , c a n be written as Nv,>c)^ - ^ L p U y . ^ • ( 7 . 5 . 6 ) where !(v,?<j/^ i s the free t r a n s l a t i o n a l state eigenf unction, see Eqs. ( 2 . 4 . 2 1 ) and ( 2 . 4 . 2 2 ) . As well,observable eigen-functions are lu<y.*>>t - - ^ L p l ^ ^ . <7.5.7) i n terms of the free observable eigenfunction,Eq. ( 2 . 4 . 2 3 ) . These eigenfunctions are orthogonal, Ju(v,x)| U ( v ' , X ' % 246 since ^ - ^ - r - ^ - T = 1. (7 . 5 . 9 ) L p L p They are complete i n the ( l - G * ) space only i f the potential superoperator does not support any bound states i n the (l-(P) space?that i s , i f j d v j d * |u(y,j)^ <^U(v,x)| = i l _ H f T = (?. - (7.5.10) L p L p O p where C? c i s the projection onto the continuum (non-discrete) eigenvalues of <3fp. Of course t l B e f a a ^ y 1 S a discrete eigenfunction of ©^p having zero frequency. The M i l l e r superoperator f o r o £ p can be written as a time i n t e g r a l , o -^- L t > = 1 - i j d s e x p ( i ^ p s ) # p e x p ( - i / f t r s ) , (7.5.11) -00 so that the state eigenfunction,Eq. (7.5.6).becomes the sum of a free part and a scattered part, + lim + (y . 2 c-/p+iV- 1^ P|(y.ic)^ r 7 247. where the scattered part i s u S G ( y . * ^ - l i r v ( y - s - < 4 + i ? r l ^ p l ( y ' * ^ • 1 ( 7 . 5 . 1 3 ) A similar result holds for the observable eigenfunction. 248. 7 . 5 . P.SEPARABLE NATURE OF l£T The L i o u v i l l e superoperator given by Eq. ( 7 . 5 . 4 ) , i s a generalized separable^ pot e n t i a l superoperator. This can be demonstrated by using Eq. (7.5.1) i n Eq. (7.5.4)> that i s , ^P = - l\f,aa^ X^tr,^ef , a a l ef,aa| + l 5ef ,aa^ "aajaa ^ V . a a • » - 5 . 1 * ) where "ia.'aa " / S e f , a a l X r K t • (7.5.15) Eq. ( 7 . 5 « l 4 ) can be written as " K f . a a ^ /(^ia.'aa-^tr ' V.aal ^ ( ^ i a . ' a a - ^ t r ' ^ f . a a ^ ^ e f . a a l -*|(t*%r-*<a>aa)\f,aJ>f / ( l + 7 r t r - * < ^ a a ) S e f , a a | . (7-5.16) 249 where the l a s t form of Eq. (7.5.16) i s the sum of two separable terms. Since the L i o u v i l l e superoperator 2 ^ i s separable,then the eigenfunctions o f c a n be e x p l i c i t l y calculated. 250 7. 5.c.EIGENFUNCTIONS Eq. (7»5»2),the eigenfunction equation,can be rewritten as where To obtain Eq. ( 7 . 5 . 1 7 ) from Eq. ( 7 . 5 . 2 ) ,use has been made of the condition (l -<?) lu(v,X)^ = l u ( y , * ) ^ , ( 7 . 5 . 1 9 ) which demonstrates that these eigenfunctions are a l l orthogonal to the @ space. Since E y ^ ( v , 2 ) appears on the rig h t hand side of Eq. ( 7 . 5 . 1 7 ) , i t i s convenient to define the scattered part of the eigenfunction.Eq. ( 7 . 5 . 1 3 ) ,as the product \ u S C < y . * ^ = l u P ( M % 4 i }<y.s>- ( 7 . 5 . 2 0 ) P The quantity U (v,?<) then s a t i s f i e s the inhomogeneous equation \T-V 2 )|»P(y.x)>^  = | l e f \ , ( 7 . 5 . 2 1 ) which i s obtained by i n s e r t i n g Eq. ( 7 . 5 . 2 0 ) i n Eq. (7 .5 . 17) 251 together with the incoming condition / r , p l u P ( y , s k — 0 , ( 7 . 5 . 2 2 ) 0-v ~ p - r — » - oo which a r i s e s from the scattering condition of. the M i l l e r superoperator. In the phase space representation,Eq. ( 7 . 5 . 2 1 ) becomes • " 5ef,aa (J'P>- ( 7 . 5 . 2 3 ) Consistent with Eq. ( 7 . 5 . 2 2 ) .the solution of Eq. ( 7 . 5 . 2 3 ) i s U^Azp+b,p) = iCn/p)Jdz fexp(i(z-z*)/iy.2c/p) «* ~ " -00 Here the pos i t i o n r has been written as the sum of a component i n the p direction,z=r«p,and a component b perpendicular to p. p In a s i m i l a r manner,the observable U (y,X) becomes, i n phase space, i ( 7 . 5 . 2 5 ) = i h 3 (^/p )/dz • exp( i (z- z • )a Y - ^ /p) B e f ^  a a (z' p+b, p). 252. where the h J f a c t o r a r i s e s from the difference i n d e f i n i t i o n s of the i d e a l phase space elements |r,pX and l r , p % .see Eq. ( 2 . 2 . 2 8 ) and ( 2 . 2 . 2 9 ) . The matrix elements,Ey^ ^ (y.ar) and E y ^ ( y , x ) .can be obtained by using the orthoganality of the state and observable eigenf unctions with the ^ s p a c e j t h a t is.using Eq. ( 7 . 5 . 1 ) and (7 .5-19) and Using Eqs. (7 -5 .12) and ( 7 . 5 . 2 0),Eq. ( 7 . 5 . 2 6 ) leads to • - i 5 e f , a a » ^ ^ ^ / / 5 e f , a a \ u P ^ ^ -( 7 . 5 . 2 8 ) S i m i l a r i l y , t h e observable matrix element i s ( 7 . 5 . 2 9 ) Depending upon the choice of B „ .these eigen-functions can be used i n the straight l i n e approximations and i n the FMR. ( 7 . 5 . 2 6 ) ( 7 . 5 . 2 7 ) 253. 7.6.DISCUSSION The time c o r r e l a t i o n function involved i n the k i n e t i c cross section has been written as the matrix element of a resolvent. Zwanzig's projection method was then used to write the relevant part of the resolvent i n terms of a memory kernel. The contribution to the exact cross section due to the resolvent was then given i n terms of matrix elements of the memory kernel. I t was shown how the expression involving the memory kernel can reproduce the o r i g i n a l projected resolvent,a r e s u l t that demonstrates the exact nature of the Zwanzig projection method. This projection method was applied to the exact quantal DWBA with the r e s u l t i n g memory kernel expression f o r the cross section being given. As well,the memory expressions f o r the semiclassical-type approximations of chapter 6 were presented. Other approximations were consid-ered. In one type of approximation,the (1-(P) projectors were neglected i n the memory kernel. As a result,the relevant part of the o r i g i n a l resolvent became proportional to the inverse of the frequency of the state operator,which i s then i n f i n i t e f o r zero frequency. This demonstrates the importance of the ) projectors to the memory kernel. On retai n i n g the ( 1 - ) projectors,a free memory resolvent approximation (FMR) to the DWBA resolvent was presented. In this,the projection of °t^T was replaced by the projection of the free generator^. r« This approx-imation scheme i s summarized i n Table 7.1. In conclusion. Kinetic Cross Sections(QM) \ KDWBA(QM) II PKDWBA(QM) PKDWDIA PKFMRSIA PKFMRDIA PKFMRCTA PKDWFMRFCA •PKDWSIA--PKDWCTA •PKDWFGA PKSLFCA=CAA • PKSLSIA PKSLCTA Table 7.1 P = Projection Formula QM = Quantum Mechanics 255. the eigenf unctions of ( l - ( ? ) / ^ . r ( l - &) f o r a one dimen-sio n a l projection were presented. The superoperator (l-CP) C^j-r( 1- ^ ) has the form of being a dynamic generator whose potential superoperator i s separable. CHAPTER 8 CONCLUSION AND DIRECTIONS FOR FUTURE WORK 257-8.CONCLUSION AND DIRECTIONS FOR FUTURE WORK A un i f i e d description of c l a s s i c a l and quantal scattering has been presented. This u n i f i c a t i o n was accom-plished by expressing a l l of the scattering phenomena i n terms of s t a t i s t i c a l states and observables. Mathematically, these are represented as phase space functions,which i s the usual c l a s s i c a l representation. Quantally the Weyl corre-spondence 1 was used to connect the Hil b e r t space operator representation of s t a t i s t i c a l states and observables to the phase space representation. Of the many equivalent phase 2 space representations ,the Weyl correspondence was chosen because,for many quantal observables,the r e s u l t i n g phase space function i s i d e n t i c a l to i t s c l a s s i c a l counterpart and because the Wigner function^ has the correct marginal p r o b a b i l i t i e s . To i l l u s t r a t e the s i m i l a r i t i e s between the two mechanics,a moment method,approximating the time dependence of the average p o s i t i o n and average momentum, was presented. In t h i s approximation scheme,the only difference between the two mechanics l i e s i n a constraint on the second moments due to the uncertainty p r i n c i p l e f o r the quantal case while c l a s s i c a l l y there i s no such condition. This moment method was compared to Heller's quantal time dependent Gaussian wavepacket approach. The difference between these two methods ( i n the quantal case) l i e s i n the equations f o r the average position and momentum, Heller** assuming c l a s s i c a l dynamics,while,in the moment method,the time derivative of the average momentum has an 258. extra term involving the t h i r d order derivative of the pote n t i a l evaluated at the average position. This extra term appears i n the c l a s s i c a l moment method as well, provided the second order moments are i n general non zero. A future project i s to see what effects t h i s term w i l l have when the moment method equations are used,instead of Hamilton's equations,to evaluate the d i f f e r e n t i a l cross section d i r e c t l y . An expression f o r the generalized d i f f e r e n t i a l cross section,Eq. (3 .4 .15) ,was developed both from the time dependent t r a j e c t o r y and stationary state beam pictures of the scattering process. This was formulated i n a mathematical language that i s applicable to both c l a s s i c a l and quantum mechanics. In doing so,the r e l a t i o n between the generalized scattering f l u x and the "ordinary'' scattering f l u x was elucidated;that is,they are simply d i f f e r e n t angular resolutions of the expectation value of the f l u x obser-vable. The average f l u x observable was written i n terms of the M/ l l e r superoperator. In the stationary state case, t h i s was related to the streamlines recently stressed by Hirschfelder and coworkers-*,which gives a very picturesque view of the scat t e r i n g event. Here,the s p e c i a l case of a c l a s s i c a l hard sphere was considered. Calculations of the streamlines (and cross sections),both c l a s s i c a l l y and quantally,for other potentials needs further study. The generalized d i f f e r e n t i a l cross section was 259. written i n terms of a time c o r r e l a t i o n function v a l i d f o r both mechanics. This led nat u r a l l y to a semiclassical-type approximation scheme where dynamic-like and s t a t i c - l i k e interference e f f e c t s were separately emphasized. In the DIA, the f u l l quantal evolution superoperator was retained while the c l a s s i c a l l i m i t of a l l other quantities was taken. This approximation has not been pursued further i n t h i s thesis because,in general,the e x p l i c i t evaluation of e x p C i ^ s ) i s unknown f o r n o n - t r i v i a l scattering potentials. One method of approaching t h i s problem,proposed by Leaf , i s to write t h i s evolution superoperator i n terms of the evolution operators expC-iH^s/h),and then to use the semiclassical Feynman path approximations of Marcus', M i l l e r 8 and Pechukas^,for each evolution operator. A more straight forward method would be to apply the Feynman path approach d i r e c t l y to the evolution superoperator and then to make semiclassical approximations to this,with, presumably,the phase point to phase point c l a s s i c a l evolution superoperator as the l i m i t . The SIA emphasized the s t a t i c - l i k e interferences i n the pot e n t i a l superoperator 2/^  while supressing the dynamic-like interference e f f e c t s by using the c l a s s i c a l phase point to phase point evolution superoperator. On using the eigenfunctions of % as a complete set of quantit-i e s f o r expansion purposes,the SIA cross section was written i n terms of c l a s s i c a l t r a j e c t o r i e s . For t h i s reason,the SIA was also c a l l e d the CTA. I f these c l a s s i c a l t r a j e c t o r i e s 260. are taken as straight lines,the quantal Born cross section r e s u l t s while taking straight l i n e t r a j e c t o r i e s i n the expression f o r the c l a s s i c a l cross section defines the c l a s s i c a l Born cross section. Other t r a j e c t o r y approximations could also he considered,in p a r t i c u l a r those "based on the moment method of chapter 2 . The r e l a t i o n between the CTA 7 3 o and the standard semiclassical approximations'' ' 7 i s yet to be elucidated. The CTA may provide a method alternate to 10 the standard work of Ford and Wheeler ,for studying the c l a s s i c a l i n f i n i t i e s (rainbows,glories and orbitings) and t h e i r associated semiclassical structures. I f the pote n t i a l i s a sum of two terms,a double poten t i a l formula f o r the generalized cross section was obtained. A semiclassical-type approximation scheme s i m i l a r to that of the single p o t e n t i a l case was presented. The quantal distorted wave Born approximation to the general-ized cross section was obtained and,by analogy,the c l a s s i c a l DWBA was defined. For both the single and double potential cases,the e f f e c t s of the i n c l u s i o n of i n t e r n a l states i s yet to be considered from t h i s viewpoint. The treatment of reactive c o l l i s i o n s also requires exploration. 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Prugovec'ki ,A Unified Treatment Of Dynamics And Scatter-ing In C l a s s i c a l And Quantum Mechanics,Preprint. 7. E.J. H e l l e r , J . Chem. Phys. 62,1544(1975);64 ,63(19?6);6$ ' 4979(1976)-66,5777(1977). 8? R.F. Snider,J. Chem. Phys. 61 ,3256(1975). 9 . D.A. Coombe,R.F. Snider And B.C. Sanctuary,J. Chem. PhysS* 61 ,3015(1975). 10. J.O. Hirschfelder.A.G. Christoph And W.E. Palke.J. Chem. Phys. 6 l ,5 k 35(1974);J.O. Hirschfelder,C.J. Goebel And L.W. Bruch,J. Chem. Phys. 61,5456(1974);J.0. Hirschfelder And K.T. Tang.J. Chem. Phys. 64,760(1976);6£,470(1976). 11. E. Prugovecki,A Quantum Mechanical Boltzmann Equation For One-Particle T - D i s t r i b u t i o n Functions,Preprint. s 12. R.E. Turner And R.F. Snider,Can. J. Phys. 34,1313(1976). 13. R.F. Snider And R.E. Turner,Can. J. Phys. 54 ,1328(1976). 14. see e.g. ,M.D. Srinivas And E. Wolf,Phys. Rev. D,l_l , l 4 ? ? (1975) . 26?. CHAPTER 4 1-. R.E. Turner And R.P. Snider,Can. J. Phys. i 4 , 1 3 1 3 ( 1 9 7 6 ) . 2. D.A. Coombe,R.F. Snider And B.C. Sanctuary,J. Chem. Phys. 61 ,3015(1975). 3 . N.F. Mott,Proc. Camb. Soc. Math. Phys. S c i . 2 £ , 5 5 3 ( 1 9 3 1 ) . 4 . see e.g.,A. Messiah.Quanturn Mechanics Vol. 2 ,(North Holland Publishing Company,Amsterdam,The Netherlands,, 1966.) 5 . see e.g.,W.H. M i l l e r , Adv. Chem. Phys. 25 ,69(1974) , R.A. Marcus,Chem. Phys. Lett. 2»525(1970);J. Chem. Phys. ^ , 3 9 6 5 ( 1 9 7 1 ) . 6. K.W. Ford And J.A. Wheeler,Ann. Phys. (NY) £ , 2 5 9 ( 1 9 5 9 ) . 268 CHAPTER 5 1. R.F. Snider And R.E. Turner,'Can. J. Phys. ^4,1328(1976). 2. J.M. Jauch.B. Misra And A.G. Gibson,Helv. Phys. Acta 4l t513( l968);R.E. Turner,J. Chem. Phys-. 6?_,5979( 1977). 3 . D.A. Coombe,R.F. Snider And B.C. Sanctuary,J. Chem. Phys. 62,3015(1975). 269. CHAPTER 6 1. R.F. Snider And B.C. Saactuary,J. Chem. Phys. 15 , 1555(1971) . 2. D.A. Coombe.R.F. Snider And B.C. Sanctuary,J. Chem. Phys. 62 ,3015(1975). 3 . F.R. McCourt And H. Moraal.Chem. Phys. Lett. 2 ,35(1971) . 4. F.M. Chen,H. Moraal And R.F. Snider,J. Chem. Phys. £2, 542(1972). 5. I. Oppenheim And M. Bloom,Can. J . Phys. 12,845(1961). 6. J.T. Hynes And J . M. Deutch,J. Chem. Phys. il , 4 ? 0 5 ( 1 9 7 0 ) . 7. R.F. Snider And R.E. Turner,Can. J . Phys. 34,1328(1976) . 8. J.M. Jauch,B. Misra And A.G. Gibson,Helv. Phys. Acta. 41 ,513(1968). 9 . R. Zwanzig,J. Chem. Phys. 11 ,1338(1960). 10. B. Leaf,J. Math. Phys. 2 , 6 5 ( 1 9 6 8 ) . 11. see f o r example:R.P. Fevnman.Statistical Mechanics A Set Of Lectures(W.A. Benjamin,Inc. Reading,Massachu-settes, 1 9 7 2 ) . 2?0e CHAPTER 7 1. R. Zwanzig.J. Chem. Phys. 12,1338(1960). 2. R.D. Levine.Quantum Mechanics Of Molecular Rate Processes (Oxford University Press,Ely House,London W.1,England,1969). 3. f o r a generalized separable potential,see e.g.jR.G. Newton. Scattering Theory Of Waves And Pa£ticJLes(McGraw-H i l l , New York, 196-67. 4. I. Oppenheim And M. Bloom,Can. J. Phys. 12,845(1961). 2?1 CHAPTER 8 1. H. Weyl.Z. Phys. 46,1(1927). 2 . see e.g. :M.D. Srinivas And E. Wolf,Phys. Rev. D,U, 1477(1975). 3 . E. Wigner,Phys. Rev. 40,479(1932). 4. E.J. H e l l e r , J . Chem. Phys. 62,1544(1975)•64,63(1976). 5. see e.g.:J.O. Hirschfelder And K.T. Tang,J. Chem. Phys. 65,470(1976) . 6. B. Leaf,J. Math. Phys. 2»769(1968). 7. R.A. Marcus,Chem. Phys. Lett. 2 , 5 2 5 ( 1 9 7 0 )1J. Chem. Phys. 54,3965(1971) . 8 . W.H. Miller,Adv. Chem. Phys. 25 ,69(1974) . 9 . P. Pechukas.Phys. Rev. 181.166(1969). 10. K.W. Ford And J.A. Wheeler,Ann. Phys. (NY) 2*259(1959). 11. I. Oppenheim And M. Bloom,Can. J . Phys. .22,845(1961). 12. R. Zwanzig,J. Chem. Phys. 22 ,1338(1961). 272. APPENDIX A:CLASSICAL CROSS SECTION The formal expression f o r the cross section given i n terms of ^ /,Eq. (3»5»3)» has previously been shown1 to give the standard c l a s s i c a l cross section, namely Crgsc(p"->R) = (bdb/sin*d%( - S(R-P " ) c r ^ t f p . . , (A.l) i n the appropriate l i m i t , and f o r f i n i t e ranged potentials. Here i t i s shown how the reduction can be done at an e a r l i e r stage, s p e c i f i c a l l y begining with Eq. (3«3«3)« While Eq;* (A.l) i s s t r i c t l y v a l i d only f o r repulsive potentials, the present treatment allows f o r multiple contributions to the cross s e c t i o n ^ Otherwise, the value of such a detailed calcu-l a t i o n i s that i t illuminates the structure of the equations i n chapter 3» Only s p h e r i c a l l y symmetric pote n t i a l scattering i s treated here. C l a s s i c a l scattering i s usually considered f o r class-i c a l l y pure states, where the dispersion i n po s i t i o n and momentum are both zero. In t h i s case, the incoming d i s t r i b u -t i o n function i s at zero time, f , M O ( r , p 0) = J(r-b) £(p-p"), (A.2) namely the l i m i t of Eq. ( 3 * 4 . 3 ) . Again i t i s required that b be perpendicular to p". From Eq. ( 3 . 2 . 4 ) i t follows that N i n c = l , while accor-ding to Eq. ( 3 . 2 . 3 ) . the incoming f l u x i s 273. j i n c ^ ) ^ = (r^)S(R-b-p -t^). (A.3) During the time that the pot e n t i a l i s acting, the tra j e c t o r y i s curved. However, i f the poten t i a l i s f i n i t e ranged, then at a pos i t i o n R outside of the range "a" of the potential, the t r a j e c t o r y i s straight, both before and a f t e r the c o l l -i s i o n . In p a r t i c u l a r , a f t e r the c o l l i s i o n , the d i s t r i b u t i o n function has the form f ( r , p ( t ) = ^ [ r - V - C p ^ K t - l j J S(p-P f). (A.4) i n which the f i n a l momentum p^ i s given i n terms of the i n i t i a l momentum p" and impact parameter b, by P^Cb.p" ) = p"costf+ p " s i n * b. (A. 5) HereT^ X(b,p" )£• -oo i s the dynamical d e f l e c t i o n angle, given 2 by the standard textbook formulae as a function of the magnitudes b and p". The f i n a l impact parameter (vector) b' i s perpendicular to p«, given e x p l i c i t l y by b* = bcos# - bsin# p", (A.6) while £ i s the time delay ( c o l l i s i o n duration time) of the c o l l i s i o n . A l l the quantities X, t ,b* and p„ are determined by the dynamics of the c o l l i s i o n with the appropriate incoming conditions contained i n the d i s t r i b u t i o n function of Eq. (A.2). 2?4. For positions R outside the range "a" of the potential, the f l u x , Eq. (3«2.3)» becomes j(R|t) = (p f ^ ) S J R-5'-(p f ^ i)(t-t ) 3 g)(t) + (P"/^) S[^-b-(p"^)t3©(-t). ( A . 7 ) The Heaviside function ® ( t ) = 1 i f t> 0 and zero otherwise, distiquishes between post c o l l i s i o n ( f i r s t term) and p r e c o l l -i s i o n (second term) contributions to the f l u x . R i s the posi-t i o n at which the observation i s to be made, thus being a dynamically independent quantity. The generalized scattered f l u x defined by Eq. (3.2.9), i s then Jgsc = [ ( P F ^ ) S[R-b'-(p f^)(t-*)J - (p"/^0 S ^ - M P / ^ H ] (H) ( t ) , ( A . 8 ) which has a contribution only f o r positive times. By Eq. (3, 2.11) the generalized number of c o l l i s i o n s per steradian i s oo j dt ^gsc -oo = R ^ p . f ® ( R . p f ) S ( 2 ) ( 5 i p f - V ) - R 2 R . p " © ( R ' P " ) S ^ ^ S l r , - - ^ ( A-9) It i s because of the simple straight l i n e nature of the 275. trajectory, that the time i n t e g r a l can be e x p l i c i t l y per-formed. Again Heaviside functions appear which require that the observer p o s i t i o n R and "scattered" momentum,p„ or p", l i e nearly i n the same d i r e c t i o n . Formally these require-ments a r i s e from the § - functions i n Eq. (A.8 ) , coupled with the Heaviside time function ( H ) ( t ) . What remains are two-dimensional § - functions f o r the components of the observer po s i t i o n R, the f i r s t being perpendicular to the f i n a l momen-tum p.p, while the l a t t e r i s perpendicular to the i n i t i a l momentum d i r e c t i o n p". By Eq. (3«3»3) the generalized cross section i s ° i e n ( P " ^ ^ ) = l i r a R 2 f d ( 2 ) b f S - P F ® ( R ' P F ) $i2)(R^ - V ) gen AJ R->O» J ~ L 1 1 * Pf ~ - R.p"©(R-p") S ( 2 )(R l p,,-b)] (A.10) This i s r e a l l y the s t a r t i n g point f o r the reduction of the generalized cross section to the form of Eq. ( A . l ) . I t i s a purely mathematical development, which i s complicated because of the number of d i f f e r e n t vector directions involved. Equation (A.10) consists of two terms, the f i r s t , a . gain term, and the second, a loss term. I f b i s outside the range "a" of the p o t e n t i a l (b>a), then p^=p" and b* =b, so the Al I At Ai A, two terms cancel. I t remains to calculate the i n d i v i d u a l terms f o r b ^ a . The loss term i s e a s i l y done, and i s considered f i r s t . In f a c t , the b i n t e g r a l i s immediate, implying that 2?6. Loss = - l i m R2R.p" ® ( R - p " h 1 gi_..| ^  a. (A.11) As R-+ *° , i t i s necessary that the p a r a l l e l component R„= R'P becomes i n f i n i t e since the perpendicular component i s bounded. Thus there are contributions only to the forward scattering since the (to) - function eliminates the backward component. As R"** , the expression i n Eq. (A.11) becomes i n f i n i t e . In order to c l a s s i f y t h i s i n f i n i t y and i d e n t i f y i t s meaning, i t i s appropriate to integrate over the allowed directions of R (subject to the constraint that b=[R2-(R.p" ) 2 J ^ a). For forward scattering, t h i s gives Loss dR = - l i m fR 2R«p"(K)(R'P")dR R*»J = - l i m 2-rrf o-ii R2cos-6-dcos-e-Jr i-< 2. X X 1 U £.T\ \ _ o i l " R J [ l - ( a / R ) 2 ] 2 1 [ l-(a/R ) 2 J * = - l i m TTR2 C O S 2-9-= - F a 2 - - C t o t . (A.12) I t follows that the loss term i s equivalent to Loss = - T a 2 S(R-P"), (A.13) which i s the second term i n Eq. ( A . l ) . The gain term has several v e c t o r i a l d i r e c t i o n s , a l l of 277. A A which must be treated with some care. Since p" and R appear i n the cross section, Eq. (A.10), i t i s appropriate to use these as a basis f o r a coordinate system. F i r s t of a l l , t h i s converted into an orthonormal system of (unit) vectors, p", n = [R-p"p"-i(I/|5-PMP"'5l = [R-P-COS«] /sin©-(A.14) A A and the t h i r d vector p" x n. The angle Q- i s the observed angle of d e f l e c t i o n , cos^=p"*R, 0 £ « - -TT , compare Eq. ( 3 . 2.7)» i n contrast to the dynamical d e f l e c t i o n angle pC(b,p" ). This coordinate system and the r e s u l t i n g c a l c u l a t i o n i s v a l i d only away from forward and/or backward scattering. In t h i s basis, important vectors are A A A R <= p"cos$- + nsin*3-% = ncos<jP + p" X n s i n <p % ' = ncos % cos <p + p'y nsin cos % -p" s i n X and zv A. A. A A p f = p"cosX +ncos(j!» sin^C +p" x nsin f sinX . (A .15) The angle 0 orients the i n i t i a l impact parameter b around the i n i t i a l momentum p", 0£<p< 2 i r . Ay c(2) To express the b -function i n terms of scalar quantities, i t i s the orientation of R and b* around p~ that i s required. For t h i s purpose, the basic set, p^, P'l= (p"-P fP f-V)/|p"-P fVp H| =(p"-p fcosX)//sin%/ (A.16) and 278. PfXPj. = P f * P " \ c s c X | . (A-.17) are employed. In t h i s basis, b' l i e s i n the plane of p„ and AJ > i p", a c t u a l l y along the +p^ d i r e c t i o n . This i s a r e f l e c t i o n of the c o l l i s i o n dynamics, which requires p",b,b' and p f to be coplanar. Which sign relates the directions of b' and p^ follows from Eq. (A.6), namely b' = -q p'i (A.18) where q i s the sign of sin%, that i s q = s i n 1/[sin%\ . (A.19) Equivalently, i f ^ r e d i s the reduced angle of def l e c t i o n , that i s 0 4 XpQ^ » which i s equivalent to X, then % * q %r e d " 2 m T 1 * ( A * 2 0 ) f o r some (zero or positive) interger m. This m parameterizes the multiple contributions to the cross section near o r b i t i n g . The component R i p of the observer p o s i t i o n R which i s perpendicular to the f i n a l momentum p f need not be i n the plane of p^ and p". On the other hand, since i t has no component i n the p^ d i r e c t i o n , i t can be written i n the form Si P f, = s-PfVs = y p i + z 5 f XPI • <A-2L) 279. From the orthonormality of the basis, the two components y and z are (the dependence on p" and R i s not stressed) y(b,<f>) = q R ( s i n ^ c o s e - s i n ^ cos % cos <p ) (A.22) and z(p) = qRsine-sin <p . (A.23) In terms of these quantities, the & 2 ^ - f u n c t i o n i s equal to 8 ( 2 ) ( 5 L p -&'> = S(y +q^) S ( z ) , (A.24) using the f a c t that b' and b have the same magnitude, namely b. An i n t e g r a l over the two dimensional impact parameter b, must be performed i n order to evaluate the gain term i n Eq. (A.10). This may be accomplished by transforming from the variables b and 9 to y and z, that i s , j d ( 2 ) b , = ^ a ° d D J 2 ( i f =Jjdydz \ J\ b p C t db d<p I = Jdy Jdz b | ^ ^ , (A .25) where J i s the Jacobian of the transformation and the ranges of the y and z i n t e g r a l s are constrained to be such that b ^ a . The gain term then becomes Gain = lim R 2 f d ( 2 ) b R-p\.® (R* Pf) § ( 2 ) ( R . -b') 280. = lim R 2((dydzb\j|R-p f.©(R-P f.) S(y +qb) S(z), , - _ A \ db ay f l = lim R 2bR-pV©(R*pV) 1 . (A.26) R-*oo 1 1 \dy dz IJ z=0,y=-qb Except possibly f o r forward scattering when &=0, the condit-ion z=0, y=-qb implies that sin<p=0. so that cp i s either 0 or IT . From Eq. (A.22), the condition y=-qb becomes % + & = -arcsin( b/R) i C() = J . (A.27) By Eq. (A.15), the inner product , 0 (A.28) i s obtained f o r the two values of the upper sign v a l i d f o r <P=0 while the lower sign applies to the case, *f = TT . J-fc follows from Eq. (A.27), that as R+ oo, % + &= nlT f o r some integer n. But by Eq. (A.28), t h i s implies that R-p f = ( - l ) n . The Heaviside function i n Eq. (A .26) thus requires n to be even. On comparison with Eq. (A.20) , i t then follows that n=-2m, q=cos<fand X r e d= "6" i n the l i m i t R-»<». In p a r t i c u l a r , i n order to contribute to the Gain;, term, the impact parameter b must be such that ^(b.p") = q £ - 2m1f. (A.29) There are i n general, several contributions, l a b e l l e d m, that 281. A . contribute to sca t t e r i n g into the R d i r e c t i o n . To complete the c a l c u l a t i o n , the derivatives dy/db = qRdsin(^.-qe )/db = qRcos(^-q &)d%/&\> > qRd^T/db (A.30) and dz/d<P = qRsin-6-cos^ — R s i n e (A .31) R -><*> are required. Combined together, the Gain term becomes Gain m bdb I , (A.32) s i n * d * which i s the standard c l a s s i c a l r e s u l t generalizing the f i r s t term of Eq. ( A . l ) . For forward scattering, Eq. (A.32) i s i n f i n -i t e corresponding to the usual glory phenomena, a more dire c t estimation of the gain term also shows t h i s . A c a l c u l a t i o n of the "ordinary" cross section may be accomplished i n a s i m i l a r manner. The scattered f l u x as defined by Eq. ( 3 . 2 . 7 ) i s j = I R-p".Rp'M« j esc -6- = n»j csc -e-S C •» A> *t This involves the unit vector n of Eq. (A.14) i n a natural 282. way. The remainder of the c a l c u l a t i o n i s s i m i l a r to that just given. F i r s t the time i n t e g r a l i s performed and then the impact parameter i n t e g r a l evaluated. The r e s u l t obtained i s exactly the same as the gain term i n Eq. (A . 3 2 ) , which i s the c l a s s i c a l evaluation of Eq. ( 3 . 3 . l ) . 283. APPENDIX A 1. R.E. Turner And R.F. Snider, Can. J. Phys. 1313(1976). 2. see e.g.,R.G. Newton.Scattering Theory Of Waves And Particles(McGraw-Hill,New York , 1 9 6 6 ) . 

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