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Feyman's quantum mechanics applied to scattering problems Dempster, John Robert Hugh 1951

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L i f 3 A? C)T< ft* C L > ( V F E Y H M A H 1 S Q U A N T U M MEO HA K.'I 0 3  A P P L I E D TO S C A T T E R I N G P R O B L E M S by JOHN ROBERT HUGH DEMPSTER A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the Department of PHYSIOS We accept this thesis as conforming to the standard required from candidates for the degree of MASTER OF ARTS* Members of the Department of Physics THE UNIVERSITY OF BRITISH COLOMBIA* April, 1951 F E Y N M A N ' S Q U A N T U M M E OH A.'N I -0 S  A P P L I E D TO S P A T T E R I N G P R O B L E M S Abstract This thesis consists of two independent parts, both of which are"applications of the quantum mechanical methods developed recently by R. P. Feynman. Part I is concerned with the non-relativistic theory, and applies Feynman!s formalism to the simple problem of the scattering of a particle by a potential field. The method and results are compared with those of the familiar Born-approx-imation. The two procedures are shown to be equivalent and to be valid under the same conditions. Feynman1s formulae are used to calculate the-first and second order terms of the scattered particle wave function, with an arbitrary scattering^ potential. Part II uses the relativistic Feynman theory, and treats:-the scattering of positrons by electrons, and of two electrons. The calculation checks the work of H..J.Bhabha and 0. Miller, who have obtained the same results by other methods. The differential cross-sections for the two scattering processes are calculated to first order, and an estimate is made.of the feasibility of an experiment to determine whether the exchange effect described by Bhabha actually occurs in positron-electron scattering. Acknowledgements The author wishes to express his appreciation to Dr. F..A..Kaempffer, under whose direction this research has been carried out, for suggesting the topic and for much helpful discussion and guidance. He is also grateful to Dr. W. Opechowski for some useful suggestions, and to several members of the Physics Department, .particularly Dr. J. B.-Warren and Mr. G-. .M..Griffiths, for discussions on the experimental aspects of the positron-electron scattering problem* Finally, he is most grateful to the National Research Council of Canada for the award of a; Bursary and a Studentship in the two years during which this research was carried out. F E I N MAM 'S Q U A N T U M . M E C H A N I C S  A P P L I E D TO S P A T T E R ING- P R O B L E M S Table of Oontents Part I. Non-Relativistic Scattering of Particles by a Potential Field 1 Introduction 1 The Born Approximation . 2 Feynman! s Formulation 5 Comparison of the Methods. 4 Explicit Solution by Feynman's Method 6 Part II. The Scattering" of Positrons by Electrons 9 Introduction 9 Notation 10 Feynman's: Formalism. 10 General Expression for the Cross**section . . . . . . . . 15 Oross-section for Positron-electron Scattering . . . . . 18 Comparison with Other Results . • 20 Feasibility of an Experiments. . . . . . . . . . . . . . 21 Appendix Ai Evaluation of. Spurs . . . . . . . . . . . . . 24 Appendix B: Electron-electron Scattering 26 Bibliography. . . . . . . . . . . . . . 27 1 F E Y N M A N * S Q U A N T U M M E O H A H 1 0 3  A P P L I E D T O S P A T T E R I N G P R O B L E M S Part I NON-RELATIVISTIC SPATTERING OF PARTICLES BY A POTENTIAL FIELD Introduction* Part I of this thesis deals with two methods of treating scattering-problems i n quantum mechanics which differ widely i n approach. One, the well-known Bona approximation, i s a perturbas-t i o n theory solution of the Schrftdinger equation. The other, described i n recent papers by Feynman, 1 , 2»5 deals directly with the solutions of the wave equation, rather than with the equation i t s e l f . In view of the fact that Feynman1s approach leads to a simplified formulation of quantum electrodynamical problems, i t seems desirable to establish a link between this procedure and the familiar Born approximation. This thesis accordingly carried out expl i c i t l y Feynman1s treatment of a simple scatter-ing problem and demonstrates the equivalence of the two methods* 1 R..P. Feynman, Rev. Mod. Phys. 20, 567, (1948). 2 R. P. Feynman, Phys. Rev. J 6 , 749, (1949), hereafter referred to as I. 5 R. P.,Feynman, Phys. Rev. J 6 , 769, (1949), hereafter referred to as II. 2 The Born Approximation. Suppose that'a p a r t i c l e of mass m and i n i t i a l momentum hk i s scattered by a potential f i e l d - V ( r ) . To apply the Born approximation, we-eet "f ( r , t ) = u ( r ) e r i u t , with <J = "hkV2m, and proceed to solve the time-independent Schrodinger equation t W k x- (2m/h")V(r)}u(r) - 0. The method i s well known, and we include hereeonly those equations which w i l l be compared d i r e c t l y with the ,Feynman solution. The wave function for the incident p a r t i c l e i s u„= e 1 K . Green's solution of the f i r s t order equation i s u,(?J = (2m/h')\G(?z,?, )V(? t )u 0(?, )d?,, (1) where the Green 1 s function G(r t ,r, ) = -(4nrlr,.-r>11 )"'e3^ r*-"*ri1 s a t i s f i e s the equation { ^ 4 k^0C?„?,)- S f c - ? , ) . (2) We write dr = dxdydzj unless otherwise specified*'- the integration i s to be taken over a l l space. The subscript 2 on V7"means that i t operates on the variables i\, and ) = ^(x l-x l)<S(y 1-y ; ) $(zx-z,). For solutions at large r x we make the approximations^ | r L - r f | ~ r z - r, • ? l / r 4 ~ r z (5) i n the exponent and denominator respectively of the Green's function.: The f i r s t and second order solutions are then: where k t ~ kr., /rx , k 3 = kr 3 / r , . 5 Feynman*a Formulation. Feynman* s method is based on- the probability amplitude for a particle to-move from one space-time point to another. This function, called the kernel, is a sum of contributions from. each possible path, where S .is the classical action along the path. Denoting the point r,,t, by 1 and r\ , t t by 2, K(2,l) is the amplitude for a particle known to be at ?, at the time t, to arrive at r*t at. the time t t . The wave function "^(2), representing the amplitude for the particle to be found at r^,t,, is then given by the expression V<2)- $K(2 f lWl)d? , . (6) Feynman shows that the kernel is that solution of, {atf/Jt,.- H^K(2,1) = ihS(2,l) (7) which vanishes for t z<t (, where H is the Hamiitonian operator for the particle (the subscript 2 indicating that-it acts on the variables 2).,. and £(2,1) = &(r^-r, )£(tt-t, )• The kernel for a: free particle is shown to be (for t1>t, ) K ( 2 , l ) = f JS -1* expfJa j ^ - ^ l UvihCt^-t.JJ l 2 h (t x -t, ) J. (8) For a perturbation theory treatment of a particle in a potential V(r), Feynman shows that the kernel may be expanded in increasing powers of: 7: K(2,l) - K e (2,l) + K ,(2,l) -- K .(2,l) * ... (9) where K ,(2,l) = -(i/h)5K 0 ( 2 , 5)V ( 5)K 0 ( 5,l)dr 3 (lOa) K ,(2,l) = (-i/li) A^K 0 (2,5.)V(5)K 0 (5,4)V(4)K 0 (4,l)dr Jd^ (10b) etc., using dr = drdt. The time integration is limited to the interval t,>t,>t, in (10a), tt>t7>ty>t, in (10b), by the condition 4 K(2,l) = G for t ^ t , . The decisive advantage of Feynman1s theory I B the inter-pretation of the above formulae in terms of successive scatter-ing. In (10a) for example we view the particle as moving freely from, 1 to 5 (amplitude K e ( 5,l) ), being scattered by the potential at 5 (amplitude -(i/h)V(5)d-tr3), and moving as a free particle again,:to 2 (K 0 (2 ,3) ). This is summed,over a l l points 5, thus giving, the. tem.:K,(2,1). The successive terms of. (9) are regarded as amplitudes for the particle to be scattered 0, 1, 2, etc. times by the potential V. Corresponding to the expansion (9) of the kernel, the perturbed wave function may-be written AK 2) - i i ( 2 ) + f,(2) + V*(2.) +• . . . (11) w h e r e a^l) is the wave function of the incident free particle, and -\f.(2) = $K 0 ( 2 , l H(l)dr, (12a) ^,(2) ^ -(i/h)5K ( ,(2,l)V(l)^(l)dr ( (12b) etc. Comparison- of the Methods. From the foregoing discussion, i t is possible to show a.close correspondence between the two methods. The similarity of equations (12b) and (l) suggests identifying G(?t,r, ) e - i w t * = -(in/2m)5Ke>(2,l)e-i^t' dt, . ( l 5 ) Operating on (1J) with {ihtyat,,* (h72m)V*Ogives on the left, by (2), ( ^ 2 m ) K + k ^ G (?,,?, ) e ~ ^ = ($/2*)!>(r^ ) e " ^ , and on the right, using, (7) with B>- (hV2m)v^, - ( ^ m ^ i h ^ l j e - ^ ' d t , = (hV2m)S(rl-?; )e- i t J t i. 5 The- equivalence of these two expressions doestnot' prove equation (13), since-an arbitrary solution of ther Schrodinger equation:, can s t i l l be added to one side* It does show however that there: i s an exact parallel between the two methods i n spite of the widee difference i n approach. Without giving a f u l l proof of equation. (13), we w i l l study the correspondence of the two methods from another point of view. The n t h order perturbed solution i s obtained by using for the kernel the f i r s t n+1 terms of the series (9)• We now seek to modify the Hamiltonian of the particle by introducing a potential function 7^(r) such that K0(2,1)+...,-+^(2,1) i s the exact'kernel for a particle moving i n the potential V n ( r ) . Applying equation (7) givess [ i h ^ t x + (hV2m)Vx*- V„(2)UKo(2,l)-»-...+Kvl(2,l)V- ih£(2,l) (14) which defines the function V„(r). A general expression for the n^ 1 , order kernel i s KA(2,1) = -(i/H) l3K 0 (2 ,5)7(5)K„ . 1 (3,l )dT } . Using the free particle form of equation (7) gives therefore [Uib/Ztx + (*V2m)^K B (2,l) = 7(2)K«.,(2,1). Thus we see that "^(2) i s given by the equation 7„(2)^K0(2,l)+...+Kw(2,l)\ - V(2){K0(2,l)+...+K^(2,l)\. Multiplying this equation by 1k(l) and integrating over dr^, we obtain Vw(2)^(2)-e... + ^(2)\ = V(2)^(2)+...^(2)V (15) From the definition of V^, the wave function ^* must satisfy the modified SchrBdinger equation containing this potential: {lXi*/dt + (nV2m)^V'V>,\^.t..+^\ = 0. 6 Using the expression (15) for Vn, this becomes-, t + (n72m)Vi K + . . = V { %+.. .+*-,\ , and subtracting the corresponding equation with n lowered by one givesP tifid/dt + (tf/2m)^\4>* = 7i.„ (l6) This is exactly the n1*1 order equation of the Born approximation. We have thus proved that the Born and Feynman solutions.are identical, provided that the same incident particle wave function "Vfe is chosen* The condition for validity of either approximation, is that the terms of the series expansion of the wave function (11) should decrease rapidly, i . e., that h M ^ W t l = 1« Since the two methods give the same result for i t is clear that they are valid under the same conditions. Explicit solution by Feynman1a method. To illustrate the equivalence of the two methoda§ we will use Feynman's formulae to calculate explicity the f i r s t and second order terms of the wave function. We set " ^ ( l ) — ei(k-r, -t^t, ) f o r t h e v & v e junction'iof the incident particle. The zero-order wave function (12a) should of course be the same. This is easily verified by direct integration using the expression (8) for the kernel. The f i r s t order term, from (12b), i s : * ( 2 ) = - T f e f r y H ) e i < e' ?' ) d ? ' d t < -co where T(r) « I (^int/ m r e ^ V S f e t g i o t d t 7 (using r'=\r,,-r,| , t — t^-t; ) is the integral on the right of equation ( l j ) , multiplied by e^u^*. Now according-to-Feynman's interpretation of the perturba-tion formulae, ^,(2) is due to particles'scattered once at 1 and then proceeding as free particles to 2. Since a free particle has a definite velocity tik/m, we should expect a contribution to T(r) only from the neighbourhood of the point t = (m/hk)r. We rearrange the exponent* and make the substitution t = (m/ftk)r(l+S): T(r) = l-^-Y U * expfeikr— — + ^ i k r ^ t ] d t ' I2irih/ ) T 8 i . rJ . - - o The coefficient of i£ in the exponent is ibkr$/(l+£). For fairly energetic particles at large distances r^ , kr»l. Hence except for the region $<*1 the exponential is a very rapidly oscillating function of £. The function (1+1=) T varies slowly (except at{j-?-l» and here the exponential oscillates with infinite frequency), and hence the integral vanishes except for very small values of \% Thus the physical reasoning of the previous paragraph is confirmed. For small we take 1+| « 1, and then, since kr»l, extend the lower limit to The integral becomes * L ? , - CO - (m/2irih)ei k r/r. * In spite of the approximations used, this result is exact. With x = l+i, p=-ia=-|-ikr, the integral is Cx"* e - P x + i a / X d x . This can be evaluated by using a table of Laplace Transforms, such as W. Magnus and F. Oberhettinger, "Special Functions of Mathemat-ical Physics" (Chelsea, New York, 1949), p. 127. The method used has the advantage of affording a physical interpretation. 8 Comparing this with the Green's function, we see that equation (lj) is verified. Applying the asymptotic approximations (3), the fir s t order wave function becomes ^ - - s k ^ y i to >ei(2"ej'?'d?' ( l 7> which agrees exactly with the Born approximation result (4). Prom (10b), the second order wave function i s : ^(2) = (-i/h)xS^K0(2,3)V(3)K0(3,l)V(l)'V'(,(l)dr,d^ - (^iAf e ^ M W ^ I )V(fy)T(|?,-?,l)V(?()eiS-?.d?1dr: / m \ * A i ( k r i - U ' f c * ) f ^ -1 Aikr 3 r = ( 2 ^ .,.rl W ^ ) ^ - - r ™ W > > Xe^ ^ O-^d^.dr; (18) using again the approximations (5). This is identical with the Born result (5). It'will be noted that the successive integrations involved in the higher order approximation do not lead to a more com-plicated time integral (as might be expected), but rather to simple products of the. same function T(r). This is due to the separation of the time dependence of equation (13) in the factor e - i ( J t l # ^e clear, in fact, that this time integral of the kernel will always give the same function provided only that the wave function can be separated in this way. Thus this integra-tion, the most difficult step in the Feynman calculation, need not be repeated in every problem. 9 Part II THE SPATTERING OF POSITRONS BY ELECTRONS Introduction. In Part II Feynman1s formalism for particles satisfying the Dirac wave equation will be applied to the typical problem of the first order scattering of two electrons. A particular case of this problem is the interaction between, a single positive energy electron and an unoccupied negative energy state ( i . e., a positron). This case is of special interest since i t makesv possible a direct verification of the hole theory of positrons. For i f the positron is really a vacant negative energy electron state, then an exchange effect will occur in the interaction which will contribute an extra term to the scattering cross-section. If however the two particles are quite different, the exchange term will be absent. This fact was pointed out by Bhabha,^ who calculated the two cross-sections. This calculation will be repeated here using Feynman1s simplified formalism. The cross-section for electron-electron scattering will also be obtained by a very slight change in the calculation and will be compared with the result given by Miller.^ 4 H.,J..Bhabha, Proc. Roy. Soc. A 154, 195, U936") 5 0. M/ller, Ann. d. Phys. 14, 531, (1932). 10 Notation. For relativistic problems the following notation is cdnven-ient: If p is a four-vector with components p^ (y«- = l , 2, J, 4 ) , then p denotes the three space components of p. We use the summation convention P^ q^ = P^ <1^  -p-qsp-q. Also, i f p^ is not a matrix, £=P^7^ is a matrix associated with the vector p, where are the four Dirac matrices The latter satisfy YrVl' + /v//< - 2 5 A „ , where S^-lt = &lx=$n=-l, and other ^ are zero. Then < w^p„ = p^, <f^=4. Note that p_*= p-p is a pure number, not a matrix. We write d^=dp,dptdp3 and d^ p = dpdp^. In particular, x^ is the four-vector r, t (we use natural units -ft" = c =1, so that x f = t ) , dr = dxdydz, and d^ r= d?dt. Finally, V= V^/2x^ = /^ /at/+|3«-V where x^ means 2/2t for/^=4 and -^Ox, -2/2y, -?3/Qz for •^ = 1, 2, 3 . Feynman's Formalism. Feynman's relativistic formalism is-essentially the same ass that described in Part I, except that the wave function is now a solution of the Dirac equation having four components, and the kernel is a four by four matrix. The Dirac equation for a particle moving in the vector and scalar potentials (times e) A^ is ( i V - m ) V " A ^ (1) and the free particle kernel satisfies (iV, - m)K+(2,l) - 15(2,1) (2) by analogy with equation (7) of Part I. The particular solution of this equation that must be chosen is (Feynman, I, p. 752) 11 K+(2,l) = £$.(2) ^ M)e'±E^'^ ) for tx>t, ^ - H ^ j ^ C l j e " 1 2 " ^ * - * ' ) for t ^ t , Here $>n are eigenfunctions of the free particle Dirac equation, and E„ the corresponding energy values. It is convenient through-out to replace the usual Hermitian conjugate by <^  ~PP* From (5) i t is seen that electrons may be propagated either forwards in time with positive energies, or backwards in time with negative energies* An electron propagating towards the past is recognized as a positron. The above choice of the kernel is exactly equivalent to the hole theory of positrons. The analogue of equation (6) of Part I is lK2)= \K+(2,l)N(lH(l)<i?V, (4) where the integration is over a ^ -dimensional surface in space-time enclosing the point 2, and N^(l) is an inward unit normal to the surface at the point 1. In particular, i f the surface consists of a l l space at a time t,<tx and a l l space at a time t'>tz, this becomes i<2) = ^ K +(2,l)p^(l)di? -$K +(2,l')/$^l ,)dry (5). Because of (5), only electron states inV^l) and positron states in ^ ( l 1 ) contribute to the integrals. If two particles are present, the amplitude that particle 0-goes from 1 to 5 while b goes from 2 to 4 (assuming no interaction) is the product (Feynman, I, p. 755) K(5,4;l,2) = K+ f c(5,l)K+ t(4,2) (6) The subscripts a and b indicate that the matrices K + operate on the wave functions of particles CL and b respectively. Matrieess with subscript a- and those with subscript b always commute. 12 Since the two particles are identical, another process is . possible involving an exchange between them. The exclusion principle requires that the amplitudes for the two processes be subtracted, giving the net amplitude K(5,4;l,2) - K(4,5;l,2) (7) When the two particles interact, (6) no longer holds. The effect of the interaction of two electrons to f i r s t order in e 1 (regarded as the exchange of one virtual quantum) is given (Feynman, II,. p. 772) by This formula can be interpreted as follows: Electron a travels? as a free particle from 1 to 5 (amplitude K + t (5»l) )» emits a« photon ( Y ^ ) , and travels from 5 to 5 (K+ 0.(5»5) )» while electron b goes from 2 to 6 (K + ( ) ( 6 , 2 ) ), absorbs the photon ( Vj,^), and goes on from 6 to 4 ( K + b ( 4 , 6 ) ). Meanwhile the photon proceeds from 5 to 6, with amplitude & +(s^). This is summed over a l l polarizations of the photon, and all points 5 a n d 6. If tj.>ttf we would say that b emits and a. absorbs the photon, but this makes: no difference in the formula. Equation (8) can describe several processes, depending on the time relations of the points 1, 2, 5> a n < i 4. Feynman repre-sents these processes by simple space-time diagrams. Thus Fig. 1 illustrates the scattering of two electrons as described by (8), together with the interfering process whose amplitude is K 1 , ) ( 4 , 5 J 1 , 2 ) . The same kernels describe the interaction of an electron with a positron simply by reversing the time Relation of points 2 and 4, as illustrated in Fig. 2. Positroiis are dis-tinguished by the direction of the arrows on their paths. 15 F i g . 1» I n t e r a c t i o n o f two e l e c t r o n s * F i g * 2. I n t e r a c t i o n o f e l e c t r o n w i t h p o s i t r o n * G e n e r a l E x p r e s s i o n f o r t h e C r o s s - s e c t i o n * , The p r i n c i p a l a i m o f t h i s t h e s i s i s t o c a l c u l a t e t h e c r o s s -s e c t i o n f o r t h e p r o c e s s i l l u s t r a t e d i n F i g . 2. We assume t h a t i n i t i a l l y ( i n t i m e ) t h e r e a r e p r e s e n t a n e l e c t r o n i n t h e s t a t e f _ ( l ) and a p o s i t r o n i n t h e s t a t e f + ( 4 ) , and t h a t f i n a l l y t h e s e p a r t i c l e s a r e f o u n d i n t h e s t a t e s g_(5) and g+(2) r e s p e c t i v e l y . T h e s e s t a t e s a r e shown i n F i g . 2. The s u b s c r i p t + w i l l be u s e d a l w a y s o n q u a n t i t i e s r e f e r r i n g t o p o s i t r o n s t a t e s , a n d t h e s u b -s c r i p t - f o r e l e c t r o n s t a t e s . I t i s n e c e s s a r y t o compute t h e m a t r i x e l e m e n t o f t h e k e r n e l 14 given by (7) and (8) for the transition-from the state f_(l)g + ( 2 ) to the-state g„(5)fV(4)» . This matrix element i s : M, = - i e ^ g_ ( 5 ) 7 + b ( 6 )V^y b /.^(s; 6)f_ ( 5).g +b ( 6)dr ydr, (9) + ie^5^ f c ( 5)I . f c ( 6 ) V ^ V^^(8-)f_ B L ( 5)g + A ( 6 ) d r f d - r . , . . after carrying out the 5-surface integrals over dr, , dr^, di^ * dr^ , such as: iK + f c ( 5 , l)p a tf - f t ( l ) d ? 1 - f _ ( 5 ) - i f + b ( 4)(i bK t t ( 4 , 6)dr;= f +i<6) according to equation (5)• The subscript sson Mt i s used because t h i 8 matrix element depends on the spins of the various states. We take as the i n i t i a l and f i n a l states:of the two part-icles the Dirac free particle wave functions f t = L 1 o t - e - 1 ^ , g t=L l :v ±e- i (l* x, ( 1 0 ) where u^and are constant spinors satisfying the Dirac equation (j> - m)u = 0 , and pj, q ± are the momentum-energy four-vectors of the particles i n each state. These solutions are normalized i n a cubic box of volume L*. The three space components of each momentum vector assume the discrete values 2fn/L (n an integer), while the fourth component varies continuously. The Fourier transform of ^(s^) i s : j + (a*) = - t f ' j k - V 1 1 ^ ^ - * * ) (2V)Vk. (11) Because of the box normalization, the integral over the three space components of k must be replaced by a sum, with dk^= 2-v/Li £ ( • £ ) , « - 2L"EVk 1e- i k( x.-^)dk^ (11a) In order to obtain a transition probability per unit time, we assume that the interaction i s "turned on" only for a f i n i t e time T. We now substitute the above expressions (10), (Ha) 15. into the matrix:element Ms and carry out the integrations- over dr r, &f6 as follows: The spaed coordinates of each point are integrated over the range -§-L' to -§L. One time coordinate (say t r ) i s integrated from 0 to T.: The other ( t 6 ) from - <*> to +00. These integrals from the f i r s t term of Mj are: = 2 » L ^ ( ( e - i F T - l ) / ( - i F ) provided p_+k - q_'= q +- k - p^^O; otherwise the integral vanishes. We define F = p.^ -t- k^- q.^ . The second term of Mj iss identical, except that f + and g_, and therefore p + and q_, are. interchanged. Thus i n the f i r s t term k = q +- p+, and i n the second term k = q +-q_. The other conditions are the same for both terms: p- + q+ - p+ - q- = o (12a) The f i r s t of these expresses the conservation of momentum, and the second defines the quantity P. The f i n a l expression for M$ i n terms of the i n i t i a l and fi n a l momenta i s : Ms= — (_iF) L V^Vv/^(il*-£ +) u-Ab -V-i,VV ( a +" f l :- ) U - V + t J = W L J ( e - i F T - l ) / ( - i F ) [ ( l ^ j V - V O ( u + ^ v + ) - (fl.t-.a-) ( v - ^ v + ) J The transition probability from the i n i t i a l to the f i n a l state iss |M $ ( = MSM<;, and since -yv~iv this isc < = ^ ^ [ ( ^ - E , ) V i v . v 4 u . ) C^VwU^V-) + ( ^ - q - ^ f f ^ ^ u ^ u , ) (^.V„v_5-V / Uv +) - (SU-E. +•) t(a.+-g.-r2Re(v:V^u.u_^u+ff+V^%. V . / v - ) ] ^ l ^ 16 This transition probability can be summed over final state spinsc and averaged over initial state spins by using the projection operatorss(2plf) (p.+m)** The result isc | n r - i ' ? i M . r ' - = ^L6ev(p.yp^q.vq,)'(8inWFnSf (15) wheree S S = (o^-p^)^Sp\(^tm^(^m)V\Sg{(a.^m)^(^m)7^] + ii*-1- )V Sp{(l- + m) Vv (£+*m)/4 SP t(s*;-~m) V* (flr-+ m)y^ (16) i - (ai--J»+.) (fl.+-fl.--) 2ReSp\(£-+m)V^(£.+m)yi>(£+tm)V^+i-m)Yvl The first term only of S would be obtained if the two inter-acting particles were not identical. The second and last terms are due to the exchange effect. The spurs occurring in \u}x are evaluated with the:help of the following relations for the spurs of the matrices 7^ and their products: Sp(Vr) = 0; S5p(V* -{h Vv) = 0; etc., for odd products; S P ( Y / T V ) = Sp(V>VWf VJ = ^ ( S V ^ V - i^S^ + i ^ i ^ i Sp(V>v^VPVVV^VT) = V s P ( V ? V v v„v-r) -SvSp(V^yy Vr-Yr) (17) + S^ SpC-y^ Vt Vr +r) -^sp(v rv ey^ y?) A T Sp(y^vWvV. )r etc., for even products. The result (see Appendix A) is 33= 52[(a.+-^)''[(p'q+)(pvqj) -v(p.p+)(q.-q+) -m^p.q.) -m* (pvq+)+2m*} + (a**<L- )""I(P--<l*)(Pvq-) + (P-q-)(P+q+) -**(P.-P+) -mx(q_-q+)+2m<^ - (a+-£-r(i+-i-);l[-2(p.q+)(p+.q_) (18) + m*[(p-qj + (p+q-)+ (q-qj +(P,P+) *• (p^ q-) + (p;q+)] -2m* t] Now | MI * is the probability for a transition in the time T * This projection operator includes the correction factor p^/m required by Feynman's normalization (I, p. 757-8). 17 from a' state characterized by the momenta p_»p+. to a state q_,q +. To obtain a d i f f e r e n t i a l s c a t t e r i n g cross-section we must f i n d the t r a n s i t i o n p r o b a b i l i t y per u n i t time* and-per- u n i t incident f l u x j summed over a l l f i n a l states i n which the p a r t i c l e s move wit h i n a • s p e c i f i e d s o l i d angle. We may-write-dQ=(JTri|M| l dadq. y (19) aq-v The incident f l u x i s J = Vr/L* where v r i s the r e l a t i v e v e l o c i t y of the c o l l i d i n g p a r t i c l e s . Because of-the momentum conser-v a t i o n c o n d i t i o n * we n e e d t o sumover the f i n a l states of only one p a r t i c l e , say the electron. We m u l t i p l y by the energy density o f - s t a t e s (dn/dq.^) and integrate over the energy dq^. The f a m i l i a r formula f o r the number of momentum states i n a box of side L, with i n a s o l i d angle d-ft, i s = ( L/2*-)' q., -ml d J l R e c a l l i n g that F = p.¥+ q + l t -p+v. -q^,, (12b) we put dq.v * dF. The cross-section becomes As usual we make use of the f a c t that the only important con-t r i b u t i o n to the i n t e g r a l i s at F^O, and that a l l . functions except sinl-g-FT/F* are slowly varying and may be taken outside the i n t e g r a l . The i n t e g r a l becomes S ( s i n ^ F T / F ^ d F - -§-TTT. We therefore have and the condition F-0 expresses the conservation of energy: 18 Orose-aection for Positron-electron Scattering. We w i l l now find an expression for dQ i n the center of mass coordinate system of the two particles. Let the two particles move i n opposite directions along the z^-axis with equal vel -ocities v and energies -p+(f = P-f= E = Vm. ( i t i s here-that we f i r s t make explicit use of the fact that p + describes a negative energy state.) Let the electron"be scattered through an angle Then p t l- p± x=0, pt>= p^--/E ^ H D^.* We apply the conservation laws (12a), (2j) p_+ q += p + r q_. Then q + l = -p.- + q-; - q-(- for i - 1,2,5 q4^ - p^ -p_v + q.¥ = q., -2E Since q;q +^ q_q. •= mx, we have q+^  = m r i t ; = m + 7 q- * » or - ±q , and therefore q_v= -q.^ = E . Finally we have Ep_. q.: - (E -m) cos0. We then get the following expressions for the scalar products of the various four-vector momenta: P-'q- - P>-"q-i- - EZ-(E1"-m'L)cos0 = mHv ^ (V^-lJcos Pjq v= P.q- •= - E M E W ' J c o s f l ^ (^-1)0036] (24) P.-E, =q--qt = -EV (aW) = -ml(2y*-l) Introducing these values into the expression (18) for S gives S =4 [[( f ^ - l f sin ief i l + 4 ( ^ - l ) c o s ^ + 2 ( r - l ) X (l*oos*£fi)} - V^5 + 4(-y-l) +("/ a-l)'(l^cos l5)} (25) - WXf-l)einxiQX\5 ^8(y 1-l)cos 1 i 6 +4(V* -lfcoe%e\] The incident flux i s J = 2v/L*= 2JE Tntf :/L , E . The cross-section * Because of Feynman's treatment of positrons, the momentum as well as the energy of a positron has the opposite sign to that of an electron following the same path. 19 dQ therefore becomes dQ - ( e V 8 E x ) d I U £ s ) . (26) To express the cross-section in terms of a laboratory-coordinate system in which the electron is initiall y at rest* we apply a Lorentz transformation. Fig. J shows, the various momenta in the two coordinate systems. We denote quantities in the laboratory system by primes. The positron and electron are Fig. 5» The transformation from center of mass to laboratory coordinates. scattered through angles 0+ and ©J respectively with the direction of the incident-positron. If we let -p'w = 2mT+m, q.'v = 2m?+m, then since 2m -1.02 Mev., T and V are very nearly the kinetic energies in Mev. of the incident positron and the scattered electron respectively. The relative velocity of the laboratory system with respect to the center of mass system is -v = - j y M / y . Hence (1-v*)'*•*-/. The Lorentz transformation equations are therefore: -m(2T+l) - R«, = (l-v^Cp^vp.,) » -m(27VL) m(27+lj = qi = (l-vl)l(q,-vq.,) - mf/-(-»"-l)cos:9J - -m/(2V+l)v -1 sind.' =q.',= q.^-mJT^ eiriQ (27) -m](2T-2v+ir-l sinG+' = q+^ - = -nn/Al sin-fl These equations give: 20 T = f-l V = (v " - l ) s in - | e ( 2 8 ) T - V = ( - V " - l ) c o s -§-6 c o t e j = -y t a n § - 0 c o t e + ' = icoifee ^29^ The r e l a t i o n b e t w e e n t h e t w o ' s c a t t e r i n g a n g l e s i s t h e r e f o r e c o t ^ c o t e ] = T + 1 (30) S i n c e t h e c r o s s - s e c t i o n i s a n a r e a p e r p e n d i c u l a r t o t h e r e l a t i v e - v e l o c i t y o f t h e two c o o r d i n a t e s y s t e m s * dQ' = dQ. S u b s t i t u t i n g - t h e a b o v e v a l u e s i n t o t h e e x p r e s s i o n f o r t h e c r o s s - s e c t i o n * and a r r a n g i n g t h e r e s u l t i n powers o f t h e e n e r g y V t r a n s f e r r e d t o t h e e l e c t r o n * we get dQ •= ^ r 0 , " d i l ( T + l ) " ; > [ ( T 4 - l ) a ( 2 T + 1 ) * V ~ * - ( T + l ) ( 8 T x+16T+7)V " ' ^ ^ + (12T +24T+13) - 4 ( 2 T + l ) V * 4 V l ] w h e r e r 0 = eYmc* i s t h e c l a s s i c a l r a d i u s o f t h e e l e c t r o n . I f t h e exchange e f f e c t i s n e g l e c t e d b y t a k i n g o n l y t h e f i r s t t e r m o f S i n e q u a t i o n (25!)» t h e c r o s s — - s e c t i o n becomes? dQo = i r ^ d i U ^ l ) ' ' [ (2T+1)-V~* - 4(T+1)V"'+2] (32) The c r o s s - s e c t i o n f o r e l e c t r o n - e l e c t r o n s c a t t e r i n g i s d e r i v e d i n / A p p e n d i x B . C o m p a r i s o n w i t h O t h e r R e s u l t s . The p r e c e d i n g v a l u e s o f t h e c r o s s - s e c t i o n s a r e i n a g r e e m e n t , u p t o a c o n s t a n t f a c t o r , w i t h t h o s e o f B h a b h a ^ and M i l l e r . 5 * (The r e s u l t s g i v e n h e r e a r e i n each- c a s e j u s t t w i c e t h o s e o f t h e e * The n o t a t i o n o f M / l l e r and B h a b h a ^ d i f f e r s f r o m t h a t u s e d h e r e . T h e i r •{ + , 0 * a r e o u r V , T h e i r V e q u a l s 2T+1 . Bhabha;? s e i s V / T . 21 other authors.) Within this factor* Bhabha's equation (15)> p. 202, i s identical with (26) and (£5) above, and Miller's equation (74), p. 568, i s equivalent to the electron-electron cross-section (B2) given i n Appendix B. Certain values given by Mott and Massey^ appear to be incorrect'. Their expressions (15) for scattering neglecting exchange and (l6) for positron-electron scattering both contain a number of errors. Feasibility of an Experiment. To f a c i l i t a t e the plotting of numerical results, we introduce as a variable € = V/T, the fraction of the kinetic energy of the incident positron that is transferred to the electron. Each of the preceding cross-sections may be written d Q . K * a | ^ - * t •(«..•) (55) where the function: <^(T,e) has the values: for scattering with no exchange effect, and ^ = 1 " ( 1 _ (2Ttl)1)(2"(2T+2)'-)* + [z^l)(?* (2^2))^ (25) " 2 \ 2 T * 1 / I T+ 1 / (2T+lJ \T+1/ for positron-electron scattering with exchange. If no attempt were made to distinguish positrons .from elec-trons i n a scattering experiment, the measurements would correspond 6 N. F. Mott and H. S. W.-Massey, "The Theory of Atomic (.':• Collisions," (Second Edition, Oxford, 1949) pp. 571-2. 22 to one of the functions 4 o ( 0 + ( j ^ ) x < U i - « ) (56) depending on whether or not the exchange'effect is present.. For electron-electron scattering, <KT,0 in (53) becomes In equations (36) to (58), we may take eT to be the kinetic energy (in Mev) of the least energetic of the two scattered particles, whether i t is a positron or an electron. Since (/)(T,e)al for most values of T and 6, the quantity gives the order of magnitude of the scattering cross-section y> an. per unit solid angle. This quantity is plotted (on a logarithmic scale) against t for several values of T in Fig. 4. In.Fig. ,5, <j>0 and <j>+ are plotted as functions of and in Fig. 6 $0 , and <f>+ are plotted for e & 0.3, a l l for several value8zof T. The relation between fc and the scattering angle is e = sinN^S = [ l + (T+l)tan«]"' where «• is the scattering angle of the particle with energy *T (<*= 6J or 91 according as *T = V or T-V). The angle for the other particle is of course the same function of 1-t. An angle scale for different values of T is shown as well as the e scale in Fig. 5« A possible experiment would consist of directing a well-collimated and reasonably monoenergetic beam of positrons onto a scattering f o i l , and then recording the scattered positrons and Fig. 4. Mognltude of the Cross-section per Unit Solid Angle. Fig. 5. ond 4, as Functions of « and a. Fig. 6. and J + as Functions of «. 25 electrons at various angles by means of counters in coincidence. The atomic electrons in the f o i l will act as free electrons provided that the energies of both particles after the collision are much larger than the binding energies involved. This sets a lower limit of say 50 to 100 kev for V and T-V. The f o i l should be of a low Z material (e.g., carbon) to keep the binding energies small, and also to reduce scattering by nuclei. The latter has the same order of magnitude as scattering by electrons, but is proportional to Z* instead of Z: (seec-Mott and Massey,^ p. 81). It is clear from Figs. 4, 5» a n j i 6 that a successful experiment, while not impossible, would require careful tech-nique. The cross-section is very small, and the difference effect to be sought for is at most points not large. The aim should be to compare the relative shapes of the experimental and theoretical curves for <^  as a function of e, rather than to makes an absolute measurement of the cross-section. This means that an experiment using the curves in Fig. 5> (i.e., distinguishing positrons from electrons) would be most likely to succeed at energies of 5 Mev and greater, for which increases as 6-»l. The two curves for T = -§,» for instance, although they differ by about 40$, are nearly parallel in the range 0.2 < « < 0.8. Unfortunately, the positrons from the most common emitters have energies under 1 Mev. Reference to Fig. 6 suggests that low energy positrons might be used by comparing the scattering of positrons with that of electrons (without distinguishing the two particles in the positron case). At T=-§- the curves for ^ Land <J>e coincide exactly, 24 while differs appreciably. This method, in contrast to that using Fig. 5» would be less advantageous at higher energies* Appendix A. Evaluation of Spurs. It is required to evaluate the spurs in the expression (16) for SJ using the relations (17) for the spurs of products of the matrices V^ . One of the spurs in the fi r s t term of S becomes Spl(p-v/*->-m) Vy(q_f^ + m)^} = 4p.vq.f ( <$>w <^ r- ^+S^£e*)+1tai'&/.v = 4(p_v.q.^+P.fkq-») +4(m*-p_-q.JV" The other factor simply has q+-» p+ replacing p_, q_. The f i r s t term of S is therefore l60q+-p J *{ p.„q_, • sq., +• (mV-p_-q_)i^l { q+„ p • q p^ + (m'-p^) 5„ \ ~ ~ . (Al) = 52(qv-p+)"+[(pjq + )^(p+.q_) + (p_p+) (q_qj -m*"(p_-q_) -m^p^qJ*2m*l The second term of S differs only in that p + and q_ are inter-changed. It is therefores 22(q*-q-)"*l(P-q+) (P*q-) ••"(P-q-KP.-qJ -mV(p_p+) -m"(q_-q+) + 2m*} • .. (A2) The spur in the last term of S can be expanded as follows: . q : x P - ? p t t r q + r S p ( y > . W f YW<r v M O + nfspCy^VW/^) +- nrq^ p.p Sp( V* Vv h V ^ ) +• mvq_N p+<r Sp( y> Vv-V. 7 v ) „ (A3) + mvq^>q^Sp(-/>y^VvV/.Vt-^) + m p.f p^ Sp( V^yPyv y r V/-V/) + axP^, q+-tSp(Vhyf VvV^VrYv) + m"p+rq+^Sp(V/.y„ y W W r V V ) To evaluate these terms we will make use of the fact that the spur of a product' is unchanged by a cyclic permutation of the matrices in the product. We first set down the results of a summation over certain indices in some of the equations (17)* 25 Sp( V> V^ Ve ) = M V V - W + *vW) * ~ 8 J x r ( a 4) sp(^v, »^ yj= - 8 V s - 52 - 16 &*pSs-c Sp(VW/--Vr V W r V ) =4(ifj<5TN - Jf-r + i e > ^ r ) - l 6 ^ f +16 S A a - , (A5b) The six coefficients of mz i n the expression (A3) can be brought by cyclic permutations into one of the following two forms, obtained from equation (A5a): Sp( W Vy V r y . V„) - 16 •= 16 «• (A6) S p C ^ y ^ V f V W ^ ) - l 6 i y ^ f > ; = i6S > f , From the preceding expressions the f i r s t term of (A3) can be calculated: Sp(V>y^Vf "Jv.v* y^y T y )^ = -8(>ie,rS?.T - ^p>Ssr - i - S p - r ^ x ) - ^ ^ ^ - ^ - l d ^ i - ^ - 1 6 S ' f r + l 6 ^ p 5 C T -16 <f*r i?s = - 3 2 5 > f f hfT (A7) The complete expression for the spur i n the third term of S is f i n a l l y obtained by substituting i n (A3). It i s : -32(pjqJ(p +-q_) + l6n~[p_q_ + p+q_+ qjq + + p :p + + p.q+ + pvq^j -32m* (A8) Since this i s of course real, the third term of S i s -32 ( q +-p + )~ " ( q+ -q - ) { -2 ( p.- q +) ( p+- q. ) " " " " r i o n ( A 9 ) + m [ p_q ^  p+'q_ +- q^ q,. +- p_-p«. +• p.- q + + p^ q,.] -2m* \ Oombination of the expressions (Al), (A2), and (A9) obtained for the three terms of S gives equation (18). 26 Appendix;B« Electron-electron Scattering. The preceding calculation can be easily adapted to- give the cross-section for electron-electron scattering. The derivation of a l l equations up to (22) i s exactly the same, except that p_, q^ are now the i n i t i a l momenta and q_, p + the f i n a l momenta, and of course.all energies are positive. We then put p_f - q v v = E, p_> * -q + 7 = /E^-nf , p ^ q^,^ = 0, and p*_ • q*_ - (E^~-m)cos0. The scalar products of the momenta are them "P--q-= P+<U * * * (>*- (/-l)co S e] = m 4 (2V , -H) PJq+* I V ' = m*(2y*-l) •= mv(2T^l) (Bl) P-P+ = q--q* = m l[vV(y %-l)cose] - mz(272-+l) The Lorentz transformation to a laboratory coordinate system i s carried out as before. V, and 7^  , which replace V and T-V i n equation (28), are the kinetic energies i n Mev of the two electrons after the col l i s i o n . We note that the scalar products P_-q_ = p+q+- are exactly the same as i n the equation (24), and that the other two pairs are interchanged and reversed i n sign. Since the latter two occur symmetrically and always squared i n the f i r s t term of S, the f i r s t term of the electron-electron cross-section i s just dQ0 with 7t replacing 7. The second term i s identical, except that 7 Z replaces 7, . The electron-electron scattering cross-section i s therefore: dQ_ =.ir* diUTXl)"'[{(2Ti-l)*" 7;* -4(T+1 )7,"' + 2) - + {(2T+l)*7j* -4(T+l)7;' + 2} - (4T^-1)(7I7T):'] s ^ d S U T * ! ) " ' [ ^ ( 2 1 + 1 ^ ( 7 , 7 ^ - (8TVI2T+3)(7,7;l)~I +-.4] (B2) 27 Bibliography 1 R. P. Feynman*. Rev. Mod. Phys* 20, 367, (1948). 2 R. P..Feynman, Phys. Rev. J6, 749, (1949). (Also referred to as I) 5 R. P..Feynman, Phys. Rev. J6, J69, (1949). (Also referred to as II) 4 H..J..-Bhabhas Proc. Roy. Soc. A;", 15-4, 195, (193<5). 5 C. Miller, Ann. d. Phys. 14, 531, (1922). 6 N.. F..Mott and H. S.W..Massey, "The Theory of Atomic Collisions," (Second Edition* Oxford, 1949). 

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