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On the Classical Limit of Phasespace Formulation of Quantum Mechanics : Entropy

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On the Classical Limit of Phasespace Formulation of Quantum Mechanics : Entropy This document is downloaded from DR‑NTU (https://dr.ntu.edu.sg) Nanyang Technological University, Singapore. On the classical limit of phasespace formulation of quantum mechanics : entropy Wang, Lipo. 1986 Wang, L. (1986). On the classical limit of phase‑space formulation of quantum mechanics: Entropy. Journal of Mathematical Physics, 27(2), 483‑487. https://hdl.handle.net/10356/94718 https://doi.org/10.1063/1.527247 © 1986 American Institute of Physics.This paper was published in Journal of Mathematical Physics and is made available as an electronic reprint (preprint) with permission of American Institute of Physics. The paper can be found at the following official URL: [http://dx.doi.org/10.1063/1.527247]. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law. Downloaded on 27 Sep 2021 07:50:25 SGT On the classical limit of phasespace formulation of quantum mechanics: Entropy Lipo Wang Citation: J. Math. Phys. 27, 483 (1986); doi: 10.1063/1.527247 View online: http://dx.doi.org/10.1063/1.527247 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v27/i2 Published by the American Institute of Physics. Related Articles Time-dependent coupled harmonic oscillators J. Math. Phys. 53, 052101 (2012) Uncertainty inequalities among frequency moments and radial expectation values: Applications to atomic systems J. Math. Phys. 53, 043512 (2012) Quantum mechanics on Laakso spaces J. Math. Phys. 53, 042102 (2012) Ewald mesh method for quantum mechanical calculations J. Chem. Phys. 136, 114112 (2012) Classical and quantum probabilities as truth values J. Math. Phys. 53, 032101 (2012) Additional information on J. Math. Phys. Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors Downloaded 17 May 2012 to 155.69.4.4. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions On the classical limit of phase-space formulation of quantum mechanics: Entropy Upo Wang Department ofPhysics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803 (Received 30 May 1985; accepted for publication 9 September 1985) The classical limits of phase-space formulation of quantum mechanics are studied. As a special example, some properties of both quantum mechanical and classical entropies are discussed in detail. I. INTRODUCTION degree of freedom so that the Hilbert space is K = L 2(R) It has been common knowledge that quantum mechan­ and phase space is eJJ = R 2. But we wish to emphasize that ics approaches classical mechanics when Planck's constant the arguments can be easily extended to the case of many approaches zero. Rigorous investigations have been carried degrees of freedom. out during the last decade by various authors. 1-4 So far the methods employed are restricted to the quantum mechanical II. THE GENERAL CLASSICAL PHASE-SPACE operator techniques and the questions considered are mainly REPRESENTATION OF QUANTUM MECHANICAL OPERATORS partition function and ensemble average. The purpose of the present work is to examine the general problem of the classi­ The mathematical form of the general question about cal limit Ii -+ 0 by means of the so-called phase-space for­ the classical phase-space representation of quantum me­ malism of quantum mechanics. With the help of the general chanical operators is stated as follows. Suppose Aand Bare results, the unsolved problem of the behavior of quantum two Hermitian operators. Find a pair of mappings, 9 and 9', mechanical entropies at the classical limit is discussed. say, on phase space, which have the following properties: The phase-space formulation of quantum mechanics 9(A) = a(q,p), (1) has found many applications, particularly in statistical me­ A chanics and quantum optics. Its basic feature is to provide a 9'(B) = b '(q, pI, (2) framework for the treatment of quantum mechanical prob­ and lems in terms of classical concepts. Following the appear­ ance of the well-known Wigner distribution function, S many Tr(A ) = JJ a(q, p)dq dp, (3) other distribution functions have been considered. For in­ stance, the antinormal-ordered (Husimi6) and the normal­ Tr(AB) = a(q, p)b '(q, p)dq dp. (4) ordered distribution (P distribution) functions,7 the anti­ JJ 8 standard-ordered (Kirkwood ) and the standard-ordered This problem was solved satisfactorily by Agarwal and 9 distribution functions. Each of those distribution functions Wolf,9 but their results were mainly presented in terms of c­ was created for a particular purpose. 10 number space, annihilation, and creation operators. which is Considering the properties of entropies, Wehrl stated, convenient for applications in quantum optics. For the sake "It is usually claimed that in the limit Ii -+ 0, the quantum of statistical mechanics and discussions in the present paper, mechanical expression tends towards the classical one, how­ we will derive the similar results in terms of phase space, i.e., ever, a rigorous proof of this is nowhere found in the litera­ q and p, language. ture."l1 In a recent paper,I2 Beretta took the first attempt at Denote the inverse mapping of 9 by 0., i.e., this question. But some weak points can be found in Beret­ 90.=0.9= 1, (5) ta's investigation, as shown in our paper. In fact, both quan­ A tum mechanical and classical entropies are singular at the A = o.[a(q, p)], (6) classical limit, however, the difference between them does then9 vanish at this limit. The paper is organized as follows. In Sec. II we briefly A= 21rliJ J a(q, p)6. (O)(q - q, p - p)dq dp, (7) review the concepts of the phase-space formalism of quan­ tum mechanics. Some useful results are derived. In Sec. III where the 6. operator is defined by the classical limit ofquantum mechanical description is con­ 6.(O)(q' - q,p' - p) = 0. [6(q' - q)t5(p' - p)]. (8) sidered. We discuss the relation between quantum mechani­ cal and classical entropies in Sec. IV. Conclusions and dis­ Explicitly it can be written as follows: cussions are presented in Sec. V. Also, in the Appendix, we 6.(O){q' _ q, p' _ p} wish to make some comments on the problems of complete classical phase-space representation of quantum kinematics = (21rli)-2f f o.(u,v) and spectral expansion in the classical limit Ii -+ 0 discussed in Ref. 12. x exp[ - i(u(q' - q~ + vIp' - P))] du dv. (9) We are going to restrict our discussion to the case of one 483 J. Math. Phys. 27 (2), February 1986 0022-2488/86/020483-05$02.50 @ 1986 American Institute of Physics 483 Downloaded 17 May 2012 to 155.69.4.4. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions TABLE I. The filter functions .0.( u.v) for the commonly used rules of associ­ Xdu dv [!l2(ili a laq,ili a lap)] ation. where the symbol (q'Jnw denotes the Weyl-symmetrized form of the = product qnpm. e.g .• (fp)w = (fp + qpq + pf)l3. !l I (iii a laq,ili a lap) xa(O')(q -q,p -pl. (20) Rule of association .o.(u.v) From Eqs. (10) and (20) it follows that Weyl pmq" --+ (q"pm)w Standard pmq" --+ qnpm exp( - iuv12li) (21) Antistandard pmq" --+ p"qm exp(iuv/Ui) where L 21 (x, y) = !l2(X, y)l!l I (x, y). (22) The inversion is A _ Letting n j ~ !lj,j = 1, 2, and using Eq. (9), we obtain the a(q,p) = Tr(Aa(Oi(q - q,p - pl). (10) following differential relation between a'Otl(q, p) and Each mapping is characterized by a so-called filter function a(O,i(q, p): !l(u,v) (Table I), which is chosen to satisfy the trivial normali­ a(O')(q,p) = L12( - ili~, - ili~) a(O')(q,p). zation condition aq ap !l(0,0) = 1. (11) (23) A The operator a (i'i)(q - q, p - p) is defined in the same fashion In particular, we choose A = p, which is the density op- as Eq. (9) with filter function erator describing the system of interest, then a(q,p) serves as if it were a classical distribution function. Conventionally n(u,v) [!l( - u, - V)]-I. (12) = b '(q,p) <iefined by Eq. (13) is called the n-equ~valence of The problem of expressing an operator in an ordered operator B, and a(q,p) defined by Eq. (10) the !l-distribu­ form according to a prescribed rule is equivalent to an appro­ tion function, which is usually denoted by P(i'i)(q,p). Thus priate mapping of the operator on phase space. the expectation value of a quantum mechanical observable B The second mapping !l' is determined by can be written in a classical form A A (B) = Tr(pB) B = IIb '(q, p)a(i'ii(q - q, p - pJdq dp, (13) with inversion = I Ib I(i'i)(q,p)P(i'i)(q,pjdqdp. (24) b '(q, p) = 21rli Tr(Ba(O)(q - q, p - pl). (14) Also the distribution thus defined satisfies the normali­ It is clear that zation condition 9 (21rli) -19'. = (15) II P(i'i)(q,pjdq dp = Tr(p) = 1. (25) Next we wish to find the relation between two different mappings!ll and !l2 say.
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