T H E C O R R E S P O N D E N C E P R I N C I P L E B E T W E E N

C L A S S I C A L A N D Q U A N T U M Q U A N T I T I E S

Y u . I . Z a p a r o v a n n y i UDC 530.145

Quest ions of the r e e s t ab l i s hm en t of a c l a s s i ca l quantity f r o m the known quantum ope ra to r a r e inves t igated in this a r t i c le , In that case when the o p e r a t o r s a r e cons t ruc ted according to the rule p roposed in a r t i c l e [15]. The exis tence of a twofold l imi t ing t ransi t ion is p roved , in which the inves t igated rule sa t i s f i es the r e q u i r e m e n t s of the co r respondence pr inciple .

In c l a s s i ca l mechan ics the specif icat ion of the coordina tes q (ql, q2 . . . . . qN) and the m o m e n t a P(Pl, P2, . . . . PN) at the instant of t ime t comple te ly de t e rmines the s ta te of a s y s t e m with N deg ree s of f r e edo m in the the sense that the value of any physical quantity A, cha rac t e r i z ing the s y s t e m under considera t ion , can be ca lcula ted as the value of a ce r t a in function A(q, p, t) [1].

In quantum mechan ics the s ta te of such a s y s t e m is comple te ly de te rmined by the specif icat ion of the wave function ~ q , t) [2] in the sense that the exper imen ta l ly m e a s u r a b l e value (A> of the quantity A at the momen t of t ime t can be calcula ted f r o m the known ~I, with the aid of the following fo rmula :

< A > = ~ ~ (q, t) 0 (A) �9 (q, t) dq, (1)

where O(A) is the ope ra to r , r ep re sen t ing the physical quantity A in quantum mechan ic s . H e r e and in what follows dq = dqldq 2 . . . . dqN, and the integrat ion over all v a r i a b l e s runs f r o m --r to + ~ .

Understanding by the co r respondence pr inciple ~{ connection of physical theor ies by means of a l i m i t - ing asympto t ic t rans i t ion with r e s p e c t to a ce r t a in c h a r a c t e r i s t i c p a r a m e t e r of ce r t a in laws into o thers [3], it is natural , on the bas i s of this pr inciple , to r equ i re the exis tence of a definite re la t ionship between the ope ra to r O(A) and the c lass ica l function A(q, p, t). In fact , in the l imi t ing t rans i t ion f r o m quantum m e - chanics to c l a s s i ca l mechan ics , the o p e r a t o r O(A) m u s t change into the c l a s s i ca l function A(q, p, t) s ince O(A) and A(q, p, t) r e p r e s e n t the s ame physical quantity. In this connection the in te r re la t ionsh ip of the o p e r a t o r s in quantum theory should, of course , i n the l imi t ing t rans i t ion lead to an analogous i n t e r r e l a t i on - ship between the co r respond ing c la s s i ca l funct ions. In other words , the ope ra to r O(A), r ep re sen t ing the configurat ion of o p e r a t o r s O(Ai), i = 1, 2 . . . . . n, symbol ica l ly wri t ten in the f o r m

0 (A) = f ( O (A,), 0 (A~_) . . . . . 0 (A, ) ) , (2)

m u s t co r r e spond in the l imit ing t rans i t ion to the s a m e configurat ion of functions, i . e . , to the function:

A (q, p, t) = f ( A , (q ,p, t), A2 (q ,p , t) . . . . . A n (q, p, t)). (3)

F u r t h e r m o r e , the co r re spondence pr inciple mus t mani fes t i t se l f in the in te r re la t ionsh ips of quanti t ies, playing a fundamental ro le in the evolution of the sys t em, the conserva t ion laws, e tc . Thus, for example , in invest igat ing the dynamics of the s y s t e m in c l a s s i ca l mechan ics we use the c l a s s i ca l Poisson b r a c k e t s :

]=, Opj Oqj Oqs Op s

In quantum mechan ics , the quantum Poisson b r a c k e t s

i (O(A).O(B) ~O(B).O(A)) (5) {o (A), O (B)} = - (

Patriee Lumumba People's Friendship University. Translated from Izvestiya Vysshikh U chebnykh Zavedenii, Fizika, No. 6, pp. 18-23, June, 1974. Original article submitted May 3, 1973.

�9 19 75 Plenum Publishh~g Corporation, 227 West 17th Street, New York, N. Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopyozg, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $15.00.

769.

play a no less impor tan t role in the d y n a m i c s o f t h e sys t em. The co r re spondence pr inc ip le r equ i re s that the ana log of the quantum Po i s son b r a c k e t (5) in c l a s s i c a l mechanics should be the c l a s s i c a l Po i s son b r a c k e t (4}.

It is quite c l e a r that the fulf i lment of the co r respondence pr inciple , in the sense indicated above, de- pends on the law used to cons t ruc t the ope ra to r O0k) for a given phys ica l quanti ty A. For example , the co r r e spondence pr inc ip le for the Po i sson b racke t s a s soc i a t ed with the rules for the cons t ruc t ion of the o p e r a t o r s accord ing to Dirac [4] is a lways sa t is f ied. However , the Di rac rule is not unique [5, 6], which leads to contradic t ions [5] a s soc ia ted with its genera l iza t ion to all poss ib le functions A (q, p, t). Upon the cons t ruc t ion of quantum o p e r a t o r s with the aid of von Neumann 's rule [7], Weyl ' s rule [8], the s y m m e t r i z a - t ion rule [9], and the o ther well known ru les [10-14], the indicated pr inc ip le is sa t i s f i ed in the l imit ~i --- 0.

In the p r e s e n t work the conditions of the l imit ing t rans i t ion are invest igated, ensur ing the fulf i lment of the co r re spondence pr inc ip le in the sense indicated above in that case when the quantum ope ra to r s a re cons t ruc ted accord ing to the rule p roposed in a r t i c l e [15].

The inves t igated rule is based on the introduction of a ce r t a in se t of quadra t ica l ly in tegrable functions of coordina tes and t ime ~oK (q, t), sa t i s fy ing the following normal iza t ion condition:

S (q, t)j2 dq = 1. (6) K

The action of the o p e r a t o r O0k) on some function U(q, t) depends on the chosen se t ~0 K and is de te rmined by the re la t ion:

i -- ((q--q')p) O(A) U(q, t ) = ( 2 ~ h ) - N S A e ( q , p , t ) e h U(q ' , t ) dq'dp. (7)

Here and in what follows, (qp) denotes the s c a l a r product of the vec to r s q and p. The functionAG(q, p, t), the so -ca l l ed genera t ing function of the o p e r a t o r O ~ ) [16], is re la ted to the c l a s s i ca l funct ionA(q, p, t) by the following in tegra l t r ans fo rma t ion :

Ao (q, p, t) = ~ ~ (~, 7, t) m (q + ~, p + 7, t) d~d~, (8) where

N t

(q,p, t) = (2=h) 2e h (qP)~ ~ (q, t)~'~(p,t) (9)

and the following F ou r i e r t r a n s f o r m is to be unders tood by ~ (p, t): N -- ~ (qp)

~ ( p , t) =S(2~h) 2 %~(q, t) e h dq. (10)

The genera t ing function AG(q, p, t) is uniquely re la ted to the ope ra to r O(A). In fact, a s s u m i n g U(q, t) =exp (iqp/ti} in Eq. (7) and mult iplying by exp{--iqp/fi}, a f t e r in tegra t ion we obtain:

t l

Aa (q, p, t) e- h (qP) - (q") O (A) e n . (11)

In con t r a s t to p rev ious ly known co r re spondence rules [4, 7-14], the co r respondence rule (7) gives, in- dependently of the fo rm of ~01~ (q, t), He rmi t i an ope ra to r s for r ea l functions A (q, p, t), which are single valued for given ~0~, and this rule guaran tees that the ave rage values (A) will be nonnegative provided A(q, p, t) -> 0 (see [15, 16]).

It is not difficult to show that the inves t igated cor respondence rule admits an inverse t r an s fo rma t io n in the sense that the c l a s s i ca l function A (q, p, t) may be found f rom the known ope ra to r O(A). In o r d e r to p rove the given a s se r t ion , we introduce into cons idera t ion the F o u r i e r t r a n s f o r m ~0 (u, v, t) of the attxi[- i a r y function (9), so that

( q , p , t ) = ( 2 f f ) - 2 N ~ ~ (I.~, tO, t) e t(~q+vp) dudv, (12)

then introducing the following definitions:

l (13) ~ - ' (u,~, , 0 = ~(u,~,t)'

~-1 (q,p, t) = (2r~)-2~ S ~-1 (u, v, t) et(uq ~-~p) duclv (14)

763

and having substituted the function (12) into re la t ion (8), a f te r multiplying by ~a -1 (u, v, t)" exp{i(q--q ' )u + i(p--p')v} and integrat ing over u, v, q, and p, we obtain

A (q, p, t) = S 9-~ (~, vl' t) AQ (q -t- ~, P + ~, t) d~d~. (15)

With the aid of Eqs. (9) and (12), it is not difficult to obtain f rom formula (14) an explici t express ion for the inverse t rans format ion ~o -t in t e r m s of the original set g0~:

S e- lh~, et(q~+2~p) dld~]. (16)

~-' (q, t,, t) = (2~:) -~' ~ j, ~ (i' - hi, t) ~, (f' + hi, t) e-'~',~ df' tc

Formula (15) in combination with (11) and (16) de te rmines the des i red function A g l, p, t) with r e spec t to the known opera tor O{A).

The following p roper t i e s of the kernel ~a -~ follow d i rec t ly f rom Eq. (16):

S 9-' (q, P, l) dqdp = l, (17)

!

S - ~- (q-O(P-V , ,. ~-i* (q, P, t) = (2~:h) -Jr e ~ ~- (~, ,], t) d~d,l. (18)

These p roper t i e s allow one to r ep re sen t ~-1 in the form of an expansion with r e spec t to an a rb i t r a ry , com- plete, o r thonormal se t of functions • ngl):

, t, ( q p ) ~ , - 1 9 - ' (q, P, t ) = (2,~h) -~' e ~- . . Cm. (t) Z* (q) 7~ (p), (19)

n , rt~

--1 where the mat r ix Cmn is Hermit ian, which is obvious, and T race (C~n) = (2~fi) N/2.

Finally, substi tuting express ion (12) into (8) and expandingA in a Tay lor s e r i e s in powers of ~ and 7/, a f te r the appropria te t ransformat ions we obtain the different ial equivalent of Eq. (8):

A~(q ,p , t ) =-~ i ~q, -~p, t A (q ,p , t ) =-~qnA(q,p,t) , (20)

A

where the explici t fo rm of the opera to r ~r follows f rom Eqs. (9) and (12) and is de termined only by the set of functions ~ ~:

0

--_ ~.~ ~ ~. (~, t) ~ ~ + ih , t e ~ (21)

Now let us discuss the conditions for the fulfi lment of the cor respondence principle . In vir tue of the rule (7) le t the opera tors O(A,)and O(A 2) cor respond to the c lass ica l functions A 1 {q, p, t ) an d A2gl, p, t). The product of these opera to rs gives the new opera to r

O(A) = O(A1).O(A~), (22)

which cor responds to a cer ta in function A gl, P, t) in vir tue of the inverse t ransformat ion (15). We a re in te res ted in the connection between the functions A~ gl, P, t), A2gl, p, t), and A(q, p, t) under the condition that the opera tors cor responding to them are re la ted by Eq. (22). F i r s t of all let us consider the re la t ion between the corresponding generat ing functions. Taking (11) into account and using (7), f rom (21) we obtain

(23)

Confining our attention in what follows to the case N=I and expanding the generating functions AIG and A2G in expression (23) in a Taylor series in powers of ~ and ~, we obtain

= ~ (-- ih)" O~A,o(q, p, t) O'A~o (q, u, t) (24) Ao(q, P, t) ,~oz" nl Op ~ Oq n

F r o m here with allowance for Eq. (20) follows the connection between the corresponding c lass ica l functions

A__ - lh #~ A A ~r (q, p, t) = |e oqop T~pA, (LP, t) 9--qnA., (q, ~, t)]~=q. ~=p. (25)

Now let the ope ra to r 0(~) be the product of seve ra l opera to rs , i . e . ,

O (A) = ~ (A,)-O (A2)-...-O (Am). (26)

764

In this case the co r r e spond ing c l a s s i c a l functions a r e re la ted by the equation ^ - i h 2 _ ~v, 1 o

~--r (q,p, t)----[e ~162 ~ fi ~q:, Ai(q,,p,, t)]ql=q, pi=p ' /=l

which follows d i rec t ly f rom Eqs. (24) and (25). Implement ing the l imit ing t rans i t ion ti -~ 0, the following r e su l t f r o m Eq. (27):

A~ (q,p, t) = A~ (q,p, t). A2.~ (q,p, t) . . . . . A~. (q,p, t),

where the following notation has been introduced:

A~(q,p, t) = ~ ~o (~, t) A (q + ~,p, l) d~,

% (q, t) = X [ ? ~ (q, t)l ~.

we obtain

(27)

(28)

(29)

(3O)

f o r m Now we may p roceed to an a r b i t r a r y configurat ion of o p e r a t o r s (2), which we may always wr i te in the

0 (A) = : ( 0 (A~), 0 (As) . . . . 0 (An)) (31)

= ~ ~ u~,. p~ [o (A,), 0 (A~) .... O(An)], (l) Pm

where l is an in tegra l n -vec to r , m = ~ l i , Pm(xl ,x2 . . . . . x n) denotes a pe rmuta t ion of m e lements (/t of l = l

the e l emen t s ~ , l 2 of the e l emen t s x 2, l n of the e l emen t s Xn), a n d U P m a r e coeff ic ients which do not depend on 1i.

Since each t e r m of the s e r i e s (31) is a spec ia l ease of the ope ra to r (26), then in the l imit ing t r a n s i - tion fi ~ 0 f rom Eq. (31) we obtain

A~ (q, p, t) = f ( Av~ (q, p, t), A2, (q, p, t) .. . . An, (q, p, t)). (32)

Thus, an a r b i t r a r y conf igurat ion of o p e r a t o r s (31) in the l imit ing t rans i t ion h --* 0 co r r e sponds to the same configurat ion of the functions A~0.

It obviously follows f rom Eqs. (29), (31), and (32) that the t rans i t ion f r o m (2) to (3) is poss ib le if and only if the following condition is fulfilled s imul taneous ly with 5 - - 0:

% (q, t) = X I ?~ (q, t) l ~ --> ~ (q). (33) K

It should be noted that in the indicated l imit ing t rans i t ion the c o m m u t a t o r of any ope ra to r s is zero. This fact p e r m i t s one to cons ide r the l imit ing t rans i t ion for quantum Poisson b racke t s , that is , for the c o m - muta to r divided by (--i5). Thus, if the o p e r a t o r s of the quanti t ies A, B, and C a re re la ted by the quantum Po i s son b r acke t (5), then for a r b i t r a r y functions, with the aid of Eq. (24), we obtain:

C o = ~ ( - - i h ) ' - I - ( O~-A~O~Ba O n A ~ 1 7 6 (34) ~,.~ n! \ Op" Oq n Oq n Op n '

which gives the following r e s u l t for 11 ~ 0 upon allowance for (20) and (21)

C~(q,.p, t) = {A~(q, p, t), B~(q,p, t)}. (35)

Finally, the fulf i lment of the l imit ing t rans i t ion condition (33) reduces re la t ion (35) to the c l a s s i ca l Po isson b r acke t (4).

Al l of the resu l t s cons ide red above can be eas i ly genera l ized to the case of N degrees of f reedom.

Thus, the co r re spondence pr inc ip le for the rule (7) for the cons t ruc t ion of o p e r a t o r s is sa t i s f ied in connection with the following double l imit ing t rans i t ion:

1) h--+O, 2) s0(q, t)---~(q). (36)

The phys ica l meaning of the condition fi --* 0 is well known. As to the l imit ing t rans i t ion c~0(q, t) ~ 5(q), accord ing to the r e su l t s of a r t i c le [16] its phys ica l meaning co r r e sponds to the r equ i r emen t for the ex i s - tence (in quantum mechanics with the rule (7) for the cons t ruc t ion of ope ra to r s ) of s ta tes of the s y s t e m with exac t ly specif ied coord ina tes .

765

The author expresses his deep gratitude to V. V. Kuryshkin for suggesting the topic and for systematic aid in the work.

L I T E R A T U R E C I T E D

1. H. Goldstein, Classical Mechanics, Addison--Wesley (1950). 2. A . S . Davydov, Quantum Mechanics [in Russian], Fizmatgiz, Moscow (1963). 3. I . V . Kuznetsov, The Correspondence Principle in Modern Physics and Its Philosophical Significance

[in Russian], Moscow (1948). 4. P . A . M . Dirac, Proc. R. Soe. Lond., Al l0 , S:61 (1926): The Principles of Quantum Mechanics,

Oxford, Clarendon Press (1947). 5. H . J . Groenwold, Physica, 12, 405 (1946). 6. J . R . ShewelI, Am. J. Phys. , 27, 16 (1959). 7. J. yon Neumann, Nachr. Akad. Wiss. G~ttingen Math.-Physik. KI., 245 (1927); Mathematical

Foundations of Quantum Mechanics [Russian translation], Nauka (1 964). 8. H. Weyl, Z. Physik, 46, 1 (1927): The Theory of Groups and Quantum Mechanics, New York (1950). 9. R . C . Tolman, The Principles of Statistical Mechanics, Oxford (1938).

10. M. Born and P. Jordan, Z. Physik, 34, 858 (1925). 11. J. Yvon, Cahiers de Phys. , 33, 25 (1948). 12. C . L . Mehta, J. Math. Phys. , 5, 677 (1964). 13. V . V . Kuryshkin, Collection oft-he Scientific Work of Graduate Students [in Russian], Vol. 1, UDN

Press , Moscow (1968), p. 243. 14. J . R . Shewell, Am. J. Phys. , 27, 16 (1959). 15. V . V . Kuryshkin, Izv. VUZ, Fiz . , No. 11, 102 (1971). 16. V. V, Kuryshkin, Ann. Inst. Henri Poincar~, 17, No. 1, 81 (1972).

766