{"@context":{"@language":"en","Affiliation":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","AggregatedSourceRepository":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","Campus":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","Creator":"http:\/\/purl.org\/dc\/terms\/creator","DateAvailable":"http:\/\/purl.org\/dc\/terms\/issued","DateIssued":"http:\/\/purl.org\/dc\/terms\/issued","Degree":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","DegreeGrantor":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","Description":"http:\/\/purl.org\/dc\/terms\/description","DigitalResourceOriginalRecord":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","FullText":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","Genre":"http:\/\/www.europeana.eu\/schemas\/edm\/hasType","IsShownAt":"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt","Language":"http:\/\/purl.org\/dc\/terms\/language","Program":"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline","Provider":"http:\/\/www.europeana.eu\/schemas\/edm\/provider","Publisher":"http:\/\/purl.org\/dc\/terms\/publisher","Rights":"http:\/\/purl.org\/dc\/terms\/rights","ScholarlyLevel":"https:\/\/open.library.ubc.ca\/terms#scholarLevel","Title":"http:\/\/purl.org\/dc\/terms\/title","Type":"http:\/\/purl.org\/dc\/terms\/type","URI":"https:\/\/open.library.ubc.ca\/terms#identifierURI","SortDate":"http:\/\/purl.org\/dc\/terms\/date"},"Affiliation":[{"@value":"Science, Faculty of","@language":"en"},{"@value":"Physics and Astronomy, Department of","@language":"en"}],"AggregatedSourceRepository":[{"@value":"DSpace","@language":"en"}],"Campus":[{"@value":"UBCV","@language":"en"}],"Creator":[{"@value":"Seagraves, Paul Henry","@language":"en"}],"DateAvailable":[{"@value":"2011-12-16T17:20:33Z","@language":"en"}],"DateIssued":[{"@value":"1964","@language":"en"}],"Degree":[{"@value":"Master of Science - MSc","@language":"en"}],"DegreeGrantor":[{"@value":"University of British Columbia","@language":"en"}],"Description":[{"@value":"A simple method is presented for writing the matrix elements of transposition operators for discrete sets of quantum numbers. A proper product of these leads to easy computation of general permutation operators. It is shown how these operators may be constructed with operators defined in angular momentum space. Results agree with Dirac for transposition of two particles of spin \u00bd and with Kaempffer for spin 1. The calculations are performed to extend the results to spin 3\/2 and 2 along with alternate representations. Special considerations are required for fermion creation and annihilation operators.","@language":"en"}],"DigitalResourceOriginalRecord":[{"@value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/39762?expand=metadata","@language":"en"}],"FullText":[{"@value":"REPRESENTATION OF PERMUTATION OPERATORS IN QUANTUM MECHANICS by PAUL HENRY SEAGRAVES B . S c . , New Mexico Ins t i tu te of Mining and Technology, 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of PHYSICS We accept t h i s thes is as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1964 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the U n i v e r s i t y of \u2022 B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that per-m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission* Department of \/ * y ? \/ ite~l+y ? \/ ? \/ f Abstract A simple method i s presented for w r i t i n g the matrix elements of t ranspos i t i on operators for d i scre te sets of quantum numbers. A proper product of these leads to easy computation of general permutation operators . It i s shown how these operators may be constructed with operators defined i n angular momentum space. Results agree with Dirac for t ranspos i t ion of two p a r t i c l e s of spin and with Kaempffer for sp in 1. The c a l c u l a t i o n s are performed to extend the r e s u l t s to sp in 3\/2 and 2 along with a l ternate representat ions . Spec ia l considerat ions are required for fermion crea t ion and a n n i h i l a t i o n operators . - i v -Acknowledgements I should l i k e to express my s incere thanks to Professor F . A . Kaempffer, for o r i g i n a l l y suggesting t h i s top ic and for providing valuable guidance at every stage of the work. I should a lso l i k e to thank the Un ivers i ty of B r i t i s h Columbia Physics Department for f i n a n c i a l a i d i n the form of a summer s t ipend . - i i i -Table of Contests Page Abstract > i i Acknowledgements iv Introduct ion . 1 I . General Permutations Operators for Discre te Sets of Quantum Numbers 5 I I . Transpos i t ion of Angular Momentum 11 I I I . Fermion and Boson Operators 18 IV. Determinants of Operators 23 Appendix A 1: Proof that (1.6) S a t i s f i e s (1.1) and (1.2) 28 Appendix A 2: Inversion of Vandermonde*s Matrix 31 Table I: Transpos i t ion Operators i n Angular Momentum Space 34 Table I I : Determinants of Operators 35 Bibl iography 36 Introduction Although permutation symmetry i s always invoked i n the quantum-mechanical d e s c r i p t i o n of systems of i d e n t i c a l p a r t i c l e s , l i t t l e i s found i n the l i t e r a t u r e on the ac tua l construct ion of permutation operators . A notable exception i s an a r t i c l e by Serber (1934), where permutation operators are constructed out of t ranspos i t ion operators for the s p e c i a l case of four objec t s . Two other types of symmetries are known. One type, the space time symmetries, can be grasped by in troduct ion of operators whose generators represent constants of the motion. For example, the generator of displacement operators i s the momentum of an objec t , the generator of r o t a t i o n i s the angular momentum and the generator of displacements i n time i s the energy (see, for example, Merzbacher 1961). The second type of symmetries deals with i n t e r n a l a t t r i b u t e s . The phys i ca l s i gn i f i cance of t h e i r generators i s not understood at a l l at present. Most attempts to understand these better use the existence of c e r t a i n gauge imparlances as a s t a r t i n g point (Yang and M i l l s 1954). Permutation symmetry i s unique i n that i t i s defined only for more than one objec t . However because of a r e l a t i o n between sp in and s t a t i s t i c s (Paul i 1940, Luders and Zumino 1958), even s ing le objects revea l through i n t e r n a l a t t r i b u t e s - 2 -faow c o l l e c t i o n s of them w i l l behave under a permutation operator. Th i s i s not c l e a r l y understood, A system ^ of n ind i s t ingu i shab le p a r t i c l e s l a b e l l e d 1, 2, 3 , . n with complete sets of quantum numbers t\\$ T2\u00bb \u2022 \u2022 \u2022 \u00bb T n i s wr i t ten ( o . i ) of = i t , . r ^ r , , : - . . J r \u201e > . \"Indist ingulshabi l i ty** of p a r t i c l e s means that i t i s not poss ib le to a f f i x any other l a b e l to a p a r t i c l e beyond a complete set r , Expression (0.1) does not take care of t h i s requirement because there are a d d i t i o n a l p a r t i c l e l a b e l s l r 2, 3 , . . . , n beyond the sets r . Since the s tate (0.1) i s . i n d i s t i n g u i s h a b l e beyond the sets r , any reorder ing or \"permutation** of the p a r t i c l e l a b e l s 1, 2, 3 , . . . , n would y i e l d a p h y s i c a l l y ind i s t ingu i shab le s t a t e . The operators that perform t h i s symmetry property are c a l l e d permutation operators . Since d i s s e c t i o n of any permutation i n t o a product of t ranspos i t ions i s always pos s ib l e , the primary task i s indeed the construct ion of t ranspos i t i on operators . One aims p a r t i c u l a r l y at a construct ion employing operators defined i n the space spanned by the a t t r i b u t e s whose p a r t i c l e l a b e l s are to be transposed. Dirac (1958) has done t h i s for t ranspos i t i on of p a r t i c l e l a b e l s attached - 3 -to two angular momenta of amount j = ^ 2 . Kaempffer (1964) has extended D i r a c ' s representat ion to the case of angular momenta of amount j \u00bb l , and has given representat ions for t ranspos i t i on of complete sets of a t t r i b u t e s i n terms of crea t ion and a n n i h i l a t i o n operators for both fermions and bosons. In the present work a simple method i s presented for w r i t i n g the matrix elements of t ranspos i t ion operators for a d i s cre t e set of quantum numbers. From t h i s representat ion , general permutation operators are e a s i l y computed. It i s then shown from another l i n e of reasoning how i n angular momentum space these operators may be constructed using operators defined i n angular momentum space. C a l c u l a t i o n s have been performed to extend work by Dlrac and Kaempffer mentioned above to include p a r t i c l e s of angular momenta 3\/2 and 2 along with a l ternate representat ions i n each case. Spec ia l cons iderat ions are required for fermion creat ion and a n n i h i l a t i o n operators . Before one can discuss any poss ib le dynamical s i g n i f i c a n c e of permutation operators , one must possess the generators of the operators . Th i s remains an as yet un-solved problem. As a f i r s t attempt i n t h i s d i r e c t i o n , computation of the determinants of the operators was c a r r i e d out and shown to y i e l d +1 or - 1 . I f the determinant of an operator i s +1, i t should possess a connection with the - 4 -i d e n t i t y as a product of i n f i n i t e s i m a l operators . The group generated by these i n f i n i t e s i m a l operators should then contain the permutations with determinant +1 as a s p e c i a l subgroup. But t h i s group i s not known or not recognized. The fo l lowing convention about l a b e l l i n g matrix elements w i l l be used. Let p.^, ( i k , . . . be the number of d i scre te values the quantum numbers 1, k , . . . may take. For example, the eigenvalues of for given eigenvalue 2 j ( j + l ) of J has = 2j+l values . Now introduce running integers a\u00b1t \u00abk\u00bb \u2022\u2022\u2022>. * \u00b0 order the poss ib le values of i , k , . . . i n a c e r t a i n sequence. The f i r s t value of i corresponds to a^l and so on by integer values to . A l l matrices w i l l be l a b e l l e d M e t c . In p a r t i c u l a r . aiak a matrix whose rows and columns are character ized by further sets of quantum numbers w i l l be wr i t ten ^axa^t^man ' With t h i s convention the matrix elements of the d i r e c t product G=A\u00aeB can be wri t ten as usual C * i \u00ab k , \u00ab m \u00ab n \" Aa\u00b1am B f f l k \u00ab a < W i * n e r 1 9 5 9 \u00bb P- 1 7 > -I , General Permutation Operators for Discre te Sets of Quantum Numbers Needed i s an operator T which has the fo l lowing propert i e s : (a) T i s s e l f adjo int and un i tary (1.1) x = T t = T\"1 so -tUt T1'-!. (b) T i s defined only i n a product space spanned by a d i r e c t product of two spaces of equal dimension. It should have the property of interchanging the order of the d i r e c t product of any two matrices i n the space. (1.2) T A \u00ae 8 T\" 1 - B \u00ae A . Thus, i f the \" f i r s t \" subspace i s associated with the quantum numbers belonging to the \" f i r s t \" p a r t i c l e and the \"second\" subspace belongs to the \"second\" p a r t i c l e , the operator T has the e f fect of t ranspos i t ion of p a r t i c l e l a b e l s . Th i s may be expressed formal ly by denoting any operator belonging to the \" f i r s t p a r t i c l e \" (1 . 3 ) F(l] - F \u00aeI and any operator belonging to the l*second\" p a r t i c l e by - 6 -(1.4) F(2)--r\u00aeF remembering the a p p l i c a t i o n of (1.2) i s (1.5) T E f i ] F ( 2 ) T \" ' - F ( D E ( 2 - J . These requirements are possessed by the fo l lowing representat ion (1.6) T \u00ae cx> then (1.9) T l ^ > = f ' ^ > retires T2\\V>-- 12\\%>*\\V> so that (1.10) f 2 = l f = * l . The proof leading to (1.6) for t ranspos i t ion operators allows an as yet a r b i t r a r y s ignj t h i s i s not a matter of convention however. Consider the eigenstate -8-|j,m)> belonging to J 2 = j ( j+1) and J3=m obtained by add i t ion of two angular momenta of amount j (say) . In p a r t i c u l a r P look at the state | 2 J p , 2 j p ^ . It i s wel l known that t h i s s tate i s symmetric and i n the usual column-vector representat ion has element 6 (with a r b i t r a r y a usua l ly chosen as a=0) i n the f i r s t p o s i t i o n , with a l l others 0 . That i s l2jpJ2jp> = lj>j'p>|\u00aeljp,jp>2--( l . l l ) where 1 and 2 are the p a r t i c l e l a b e l s . The element T ^ ^ of T i s the only non-zero element i n the f i r s t column of T and hence i s the only element to a f fec t i 2 J 0 \u00bb 2 J \u00ab ^ and must be chosen as +1 to leave the s tate unchanged by e r o o 0 * \u2014 o * 0 e o o * \u2022 O transpos i t ion of p a r t i c l e l a b e l s . That i s \u2014 Tl2j P j2j p>=f+) | . . . o \u2022 * \u2022 e 0 e 0 o \u2022 \u2022 \u2022 o o (1.12) \u2022 \u2022 \" \u00ab * * \u2022 \u2022 \u2022 \u2022 * o \u2022 . \u2022 \u2022 o O - \u2014 Hence the s ign i s p o s i t i v e throughout for angular momenta. The representat ion (1.6) can be extended to more than two p a r t i c l e s by in troduct ion of the i d e n t i t y for unaffected p a r t i c l e s . Thus the three poss ible t ranspos i t ion operators for three p a r t i c l e s are (1.13) . {Tl%)(j.ldi^c,^c^dp a ^ . ^ J ^ ^ ' J It i s important to keep track of the subscr ipts and the spaces to which they belong. Here \" i ,m\" belong to sub-space \" i , a \" , i . e . to the f i r s t p a r t i c l e and so on. By su i tab le products of t ranspos i t i ons , general per -mutations are found. For example, f or three objects the permutation T^23 a n d T132 a r e f o u n d a s fol lows (note that the operators are ef fected from the r i g h t ) (1.14) < T . Z 3 > W , , c ^ < , ^ ** s t (1.15) (T ?) - T \" ( T 2 3 ) \/ T J ~ 2 \u2014 ^d.\/cf r \u00bb d > c f , 4\/ ^ oL oL , S t -10-The remaining permutation of t h r e e - p a r t i c l e labe l s i s the i d e n t i t y , g iv ing s i x i n a l l , I , T 1 2 , T 1 3 \u00bb T 2 3 \u00bb T 1 2 3 \u00bb a n d T132< This process may proceed to any number of p a r t i c l e s without d i f f i c u l t y . For example the permutation T 1 4 g 3 2 can be wri t ten as T 1 3 2 T 1 4 5 \u00bb From (1.14) and (1.15) i t fol lows that (1.16) ( T j j g l ^ ^ ^ ^ ^ ^ a ' ^ j r f j = ^^^^JcC.df^^d^J (1.17) ( ^ 4 ^ ^ ^oT.dj^K^fiCp.oC^p z Sd.i*pJ\u00ab3*\/*h*\u00bb(J4;*\/*J'i:\u00bb so that r I J (1.18) ( T , 4 5 3 2 J ^ d i d ^ c ^ t f ^ ^ As a mnemonic a i d , i t may be noted the column l a b e l , aj for example, of any given symbol $ccady \u00bb appears always i n conjunction with a row l a b e l , a a i n t h i s example, which i n the l a b e l l i n g of the symbol T have the respect ive p o s i -t ions required by the permutation of p a r t i c l e s with quantum numbers P and a i n t h i s case. Thus i f the permutation i s (21) as i n (1.18) , i s found with a& among the subscr ipts of the corresponding J-symbol . -11-I I . Transpos i t ion of Angular Momentum Necessary propert ies for t ranspos i t ion operators i n angular-momentum space are found by examining the e igen-states for two p a r t i c l e s of given angular momentum. P a r t i -c u l a r l y needed are symmetry propert ies under exchange of p a r t i c l e l abe l s for i d e n t i c a l p a r t i c l e s . The elgenstate | J a > ^b\u00bb \u2122y 1 D e l o n g i n S t o J 2 = j ( j + D and J 3 - m obtained by add i t i on of two angular momenta j a and j b , have the we l l known symmetry property (Condon and Short ley , 1963) If the p a r t i c l e s have the same angular momentum j p the mul t ip l e t j = 2 j p i s symmetric, the mul t ip l e t j = 2 j p - l i s antisymmetric, and so on a l t e r n a t i n g l y to j=0. Thus the needed property of the t ranspos i t ion operator T i s (2.2) t \/ ; , \u00bb > > ( - i ; 2 j p \" J l j > > . Now suppose there i s an operator F which i n the state j j , m ) has the eigenvalue f . . Further assume the f . are a l l d i f f e r e n t so that there are 2 j p + l values of f^. Then T can be wri t ten as a power ser i e s i n F , -12-(2.3) T = X0 + YIF +X2 FZ + X3F3 + - \" + X2jpF with unknown c o e f f i c i e n t s X Q , X^, X 2 , . . . to be defined so that the eigenvalues t . of T must s a t i s f y (2.4) t j ^ x 0 + f ; + y t + m \/ \u2022 2 i p i n accordance with (2 .2) . This system of 2j +1 equations form, (2.5) 2Ja ]2 *2\u00bb X 3' ... can be put ' o 4*2J> 11 x , \u2022 \u2022 \u2022 \u2022 * # 1 \u2022C2if> \\ -The c o e f f i c i e n t matrix i s a Vandermonde's matrix (see, for example, Browne, 1958)\\ the inverse of which can be wri t ten as (see Appendix A 2) -13-where the fo l lowing notat ion has been used: t p _ sum of a l l d i f f e r e n t products formed with s values f^ (2.7) I {U-f.)(iri,)(ft-f2). \u2022 - Cf,-i2jp) and where the eigenvalue f t Is excluded i n the formation of products i n the numerator and \u00a3^.-\u00a3^ i s excluded from the denominator. The s o l u t i o n can now be wri t ten (2.8) X k H - \u00bb l j r K T ( \" f i r i J j \u2022 W* Z ' - \u00ab , V . * f ? . L = 0 -14-Suitable operators F remain to be found. C l e a r l y J with eigenvalues j ( j + l ) w i l l work. If J i s wri t ten as the sum of the sp in vectors J ( l ) and j (2 ) for the f i r s t and second p a r t i c l e s re spec t ive ly (2.9) J r J ( | ) 4 J (2 ) i t fol lows that (2.10) J 2 = j ' l D ' f W * \u2022!<\u00bb.\u2022 The eigenvalues gj of J ( 1 ) . J ( 2 ) s a t i s f y then the equation (2.11) j(j>;) = jP(jP^) i-Jp(jp-hl) + 2 jj^ A l t e r n a t i v e l y then, one may use a representat ion with J (1 ) . J (2 ) for F with the property (2.12) J(l)-I(?) ft\/j,\") w h e r e 'lj=-j,1fr0 + ij(j+l). 2 2 - 1 9. If J i s d iagonal ized such that VJ V A \u00bb J ^ f the eigenvalues 2 j ( j + l ) the appear along the diagonal and J~j can be factored as (2.13) j \/ = ^(Kj+I) 15 where K = j lj>> which i s another a l t e r n a t i v e for F . Results with J(1) .J(2) agree with Dirac (1958) for sp in V2 and Kaempffer (1964) for sp in 1. These and further r e s u l t s f or T are c o l l e c t e d i n Table I . An exponential representat ion of these operators should e x i s t . One would l i k e a matrix A such that T may be wri t ten (2,16) where the fac tor i T T i s introduced for convenience. The eigenvalues a . of A s a t i s f y (2.14) J2- K where itrA 1 - \\ itr\u00ab i- \\ (2.17) -16 In accordance with (2.2) one has (2.18) ( - \/ ) 2 J p \" J = c ^ \u00ab j * * \u00ab ^ \" \u00ab j * \u00bb ( - l ) a J because the eigenvalues of T are r e a l , r e q u i r i n g a to be in tegers . Indeed A can be found by much the same procedure jus t followed to f i n d T . That i s , A may be wri t ten as a power s er i e s (2.X9) A - Y0 + Y, F + Y, F 2 + Y3 F3+ \u2022 \u2022 \u2022 + Yl}f F2'f with the unknown c o e f f i c i e n t s Y Q , Y^, Y 2 , . . . d e f i n e d so that the eigenvalues a s a t i s f y (2.20) = y o + Y, fj f Y2 *J + Y3 *j + \u2022 \u2022 \u2022 * V2j.\/\/J> where a^ i s a su i tab le even or odd integer to agree with (2.18) . This system of 2j^+l equations with 2 j p + l unknowns Yq\u00bb Y^, Y 2 , . . . can be solved i n the same manner as the system of equation (2 .4) . It i s seen that A i s not unique; the general form would be ;ir(A + B) lit-A (2.21) T = e - e where B i s any matrix with even integer eigenvalues, but -17-a l l the representat ions are equivalent and t h i s i s wri t ten i n (2.21). Rather than work out these power s e r i e s , which are not unique for a given s p i n , one would l i k e a c losed form which would always work. Using the operator K of (2.15), T may be wri t ten inr ( H(0+ he(2) - K) . p ' > ^ 2 J P \" ^ ) (2.22) T = e - c for one has (2.23) i n accordance with (2 .2) . -18 -I I I . Fermion and Boson Operators The representat ion T proposed by (1,6) cannot be appl ied as i t stands to spaces spanned by fermion creat ion and a n n i h i l a t i o n operators , because fermion states require the s ign convention of Jordan and Wigner (1928) which has not been taken care of by (1 .6) . This s ign convention i s i n t r o -duced such that a + CfJ I . . . j . . . > = \/ . . . ; 1^} .. .> (3.1) where m-t (-*-\/ T j etc. which i s not the property assumed, in. (1 .2) . The construct ion of the operator T 1 2 i n terms of crea t ion and a n n i h i l a t i o n operators for fermions i s r e l a t i v e l y easy, because there i s only a f i n i t e number of b i l i n e a r combinations containing the operators a ( l ) , a + ( l ) , a (2) , and a + (2 ) from which one can construct the most general un i tary operator invo lv ing a ( l ) , a + ( l ) , a (2) , and a + (2 ) of which T 1 2 , i f i t ex i s t s at a l l , must be a s p e c i a l case. One f inds (see Kaempffer, 1964) 0 . 7 ) T,t.-ie'<*>\u00ab\"el | T , t H One f inds the + I -far1 ir) at* t t^a. \u00bb (hi) remembering \"d\" i s the dimension of the subspace of a s ing le p a r t i c l e (d=2j p+l for s p i n ) , and the plus s ign must be chosen i n the representat ion (1.6) for sp in space. Results are c o l l e c t e d i n Table I I . -27-From (3.9) i t fol lows that the determinant of fermion t ranspos i t ion operators are +1. From (4.3) i t fol lows that for t w o \u00bb p a r t i c l e s ta tes , the t ranspos i t i on operators for p a r t i c l e s of sp in 1\/2, 5\/2, 9\/2, . . . have | T | =\u00bb - 1 . Such p a r t i c l e s cannot be i d e n t i f i e d as fermions unless they have further a t t r i b u t e s so that the t o t a l t ranspos i t ion operator has determinant +1. 2 8-Appendix A 1: Proof that (1.6) s a t i s f i e s (1.1) and (1.2) By interchanging i n the row and column labe l s one has and a lso (A 1.2) ( T l , ^ ^ ^ - ! ! < k . \u00ab A \u00ab f V A \u00ab \u00bb : t.e., T=T-\u00ab. Thus r e l a t i o n s (1.1) are v e r i f i e d . Furhter i t fol lows that (A 1.3) tI*F)t{t ^ ' ^ F ^ so that (A 1.4) \\Til*F%t.t J - ^ A * P ^ < - . ^ n = . ^ K \u00ab C m ^ < C ' \u00bb ocpocr and a l so o r . (A 1.6) T F ( 2 ) T \" 1 n F ( l ) . -29-Then (A 1.7) TTF(2)TT * TF(1)T so that (A 1.8) T F d J T \" 1 \u00bb F(2) For another operator E i t fol lows that (A 1.9) TECIJT\"\" 1, = E(2) so from (A 1.6) (A 1.10) TE(1)T\" 1 TF(2)T\"\" 1 E(2)F(1) which gives (1.2) or (1 .5) , (A 1.11) TE(1)F(2)T~ 1 = F(1)E(2) as r e q u i r e d . Assume there i s another operator S with the propert ies (1.1) and (1 .2)j that i s , (A 1.12) S = S^-: S\" 1 , SE(1)F(2)S\" 1 F (1 )E(2) . It- .follows that (A 1.13) TSE(1)F(2)ST > TF(1)E(2)T - E(1)F(2) or (A 1.14) TSE(1)F(2) \u00bb E(1)F(2)TS. -30-That i s TS commutes with a l l E(1)F(2) and hence must be a mult ip le of the i d e n t i t y (Wigner 1959, p. 75). Hence (A 1,15) TS \u00ab* n l , n a constant and thus (A 1.16) T = nS, T 2 =n 2 S 2 =I=n 2 I , and n=+l. The uniqueness except for s ign i s thus proved. -31-Appendix A 2: Inversion of Vandermonde's Matrix Vandermonde's matrix V has the form (A 2.1) i o k 2 a, where a \u00a3 i s the number a^ to the power h . The notat ion *R i s given by (A 2 2) R = SU** \u00b0* a^ ^ l ^ e r e h * products wiH 5 oi tk< yuluta \u2014 J r , The denominator i s common to a l l ^R. I f i \u00a3 j then a \u00b1 i s used i n formation of the numerators of and the product s a n - s J R c a n b e wr i t ten i n the form i s -33-(A 2.8) \u2022 \u00ab ^ I R = j , * p f \u201e \u00ab-S j , ^ where ^ f i R means exc lus ion of both i and j information of s the numerator but s t i l l only a j\"\" a j i s l e f t out of the denominator. Now i t i s seen that C J J : \u00bb 0 when i ^ j . I f i = j , one should have c^j \u00bb 1. The denominator of C j j has the fac tors a j - a s with a^'-j* a g , so i t i s s u f f i -c i ent to show that the numerator fac tors the same way. Subs t i tu t ion of root a a g \/ a^ then should y i e l d zero as has already been shown above. Thus c i j = d~ij a n d the proposed invers ion i s v e r i f i e d . Table I . Transpos i t ion Operators i n Angular Momentum Space J p F T z. Sz K O, 2 2K-I 1 1(0-1(2) J 2 K ZK2-4K+I 3 Z J(0'J(2) t . K -li . 3 3 4 , 4 , 4 , 4 O, 2. ,6,12. 0, 1, 2, 3 Z J 2 K 0 , 2 , * , \/ 2 , 2 0 0, ' , 2 , 3 , 4 -35-Table I I . Determinants of Operators (a) A l l even permutations have determinant +1. (b) Determinant of odd permutations are number of p a r t i c l e s two many s ign of eq. (1.6) (+) (+) (-) dimension d d * 2 a p + l d=2j +1 P 2 -1 -1 +1 +1 3 -1 +1 -1 +1 4 +1 +1 +1 +1 5 +1 -1 +1 -1 6 -1 -1 +1 +1 7 -1 +1 -1 +1 8 +1 +1 +1 +1 9 +1 -1 +1 -1 10 -1 -1 +1 +1 Bibl iography Browne, E . T . Determinants and Matr ices , Univ . of North C a r o l i n a Press (1958), p . 78. Condon, E . U . and G . H . Shortley The Theory of Atomic Spectra, Cambridge Univ . Press , New York and London (1963), Chapter I I I . Dixac , P . A . M . C Quantum Mechanics, 4th e d . , ^58, Oxford Univ . Press , London and New York (1958). Jordan, P. and E . P . Wigner Z . Phys ik , 47, 631 (1928). Kaempffer, F . A . Concepts i n Quantum Mechanics, Section 27, Academic Press , New York (1964). Luders , G. and B . Zumino Phys .Rev. , 110. 1450 (1958). Merzbacher, E . Quantum Mechanics) John Wiley and Sons, Inc. New York and London (1961); pp. 363-365. Serber, R. Phys .Rev. , 45, 461 (1934). Wigner, E . P . Group Theory, Eng l i sh t r a n s . , Academic Press New York and London (1959), Yang, C . N . and R . L . M i l l s Phys .Rev. , 96, 191 (1954)* ","@language":"en"}],"Genre":[{"@value":"Thesis\/Dissertation","@language":"en"}],"IsShownAt":[{"@value":"10.14288\/1.0105217","@language":"en"}],"Language":[{"@value":"eng","@language":"en"}],"Program":[{"@value":"Physics","@language":"en"}],"Provider":[{"@value":"Vancouver : University of British Columbia Library","@language":"en"}],"Publisher":[{"@value":"University of British Columbia","@language":"en"}],"Rights":[{"@value":"For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https:\/\/open.library.ubc.ca\/terms_of_use.","@language":"en"}],"ScholarlyLevel":[{"@value":"Graduate","@language":"en"}],"Title":[{"@value":"Representation of permutation operators in quantum mechanics","@language":"en"}],"Type":[{"@value":"Text","@language":"en"}],"URI":[{"@value":"http:\/\/hdl.handle.net\/2429\/39762","@language":"en"}],"SortDate":[{"@value":"1964-12-31 AD","@language":"en"}],"@id":"doi:10.14288\/1.0105217"}