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

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.

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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.

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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)

484 J. Math. Phys .• Vol. 27. No.2. February 1986 LipoWang 484

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 which has the important property that it is non-negative everywhere in phase space. 16 The class of non-negative {R (.0 )cl = f f P(CI)(q,p)R (b(cl)(q,p))dq dp, (37) quantum distribution functions has belen shown to be rather and its existence has been assumed. smal1. 17

III. CLASSICAL LIMIT Ii -+ 0 IV. RELATION BETWEEN QUANTUM MECHANICAL AND CLASSICAL ENTROPY With the help of the formulas mentioned in the Sec. II, we now consider taking the classical limit Ii -- O. Traditionally entropy is introduced in the phenomeno First of all we observe from Eqs. (11), (22), and (23) that logical thermodynamical considerations based on the sec any phase-space distribution function that describes the ond law of thermodynamics. The conception ofentropy thus same system, approaches the same limit at Ii __ 0. 18 Also, defined frequently leads to some obscure ideals. 19 The well any phase-space equivalence (resulting from any rule ofasso known heat death provides a good example. In classical sta ciation) of the same quantum mechanical operator ap tistical mechanics the Boltzmann and the Gibbs entropies proaches the same limit at Ii -- O. Explicitly we have are not very good ones either. The reason is that they never lim b '(O)(q,p) = b (cl)(q,p), (30) lead to the third law of thermOdynamics. Thus a correct 111 ...... 0 definition of entropy is only possible in the framework of and quantum mechanics. If a system is described by a density operator p, its en· (31) tropy is then defined quantum mechanically by

Of course the necessary and sufficient condition for any of S(p)= -kTr(plnp) Eqs. (30) and (31) to be true is that the appropriate limit exists, which will be assumed in the following discussions. = - k f f P(Wi(q,p)(lnp)(W)dqdp Equation (31) thus defines a classical distribution. p(CI)(q,p). We can prove thatP(C')(q,p) is real and non-riega = - k f f P (W'(q, pl( In ( 2';.,,)) (W'dq dp - k In(21T'1i). tive everywhere in phase space simply by choosing a real and non-negative quantum distribution function, e.g., the anti (38) standard-ordered distribution function, on the left-hand side Noticing that of Eq. (31). In the case of a canonical ensemble, P(CI)(q,p) (pI21rli)(w) P(W)(q,p), turns out to be the Maxwell-Boltzmann distribution.s = (39) Now we would like to consider the properties that the we find, according to Eq. (36). that the first term in Eq. (38) "classical functions" b (CI)(q, p) and P(cl)(q, p) possess. The approaches conclusions at which we just arrived make it enough to re strict ourselves within the Wigner equivalence and distribu -kff P(cl)(q,p)lnP(cl)(q,pjdqdp (40) tion function. By u~ing Eqs. (28) and (29) we get the Wigner equiv in the limit Ii -+ O. But the second term diverges to positive alence of B n, where n is a positive integer, infinity. If the classical entropy functional is defined by A .... A (BR)(W) = b '(W)(q,p)exp(IiG 121)(B n-I)(W) S(cl)(P(CI)) = - k f f p(CI)(q, p)ln P(cl'(q, p)dq dp - k In(21rli), = b ,(W)(q,p)exp(IiG 121) (41) X [b '(W)(q,p)exp(1iG 12i)(.on 2)(wl]. (32) then the quantum mechanical entropy approaches the classi Obviously, exp(1iG 12i) approaches its identity at Ii -+ O. cal entropy functional in the limit Ii -+ 0, in the following Hence sense:

(33) lim(S(p)-ln(21rli)) = lim(S(cl)(P(cI))-ln(21rli)), (42) 111 ...... 0 111 ...... 0 It is easy to see that for any infinitely diWerentiable func or tion R (t ), we have the following useful relation: lim(S (,0) - S(cl'(p(Cl,)) = o. (43) lim (R (B) )(0) = R (b (e1) (q,b»), (34) 111 ...... 0 111 ...... 0 Let us consider a simple example, i.e., an ensemble of where suffix n denotes an arbitrary n-equivalence. harmonic oscillators with a heat bath of temperature T. The On the other hand we have easiest way to compute the quantum mechanical entropy is to use the Wigner phase-space equivalence and distribution Tr(,oR (.0)) = f f P (O)(q, p)(R (.0 j)<°)dq dp. (35) function. From Ref. 20, Let Ii -+ 0 at both sides b ,(w'(q, p) = (exp( _ PH ))'W) A A lim Tr(,oR (B)) = (R (B )Cl' (36) 111 ...... 0 = sech(wP 12)exp( - 2 tanh(wp 12)Hlw), where (44)

485 J. Math. Phys., Vol. 27, No.2, February 1986 LipoWang 485

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 and the partition function is resentation of quantum kinematics for systems with both A Z = Tr(exp( -/3H)) classical and quantum mechanical descriptions. With help of the general considerations of phase-space representation = (21rli)-J fb '(W)(q,p)dq dp we wish to make some comments on Beretta's ideals and derivations. In order to keep consistent with the notations = (2 sinh(w/3 12))-1. (45) that we have been using, we quote those conditions in a slightly different form. Hence the Wigner distribution function is an immediate Given a system with quantum mechanical Hilbert space result ofEqs. (44) and (45), K and~lassical space 9, find two mappings w(q, p; p) and P(w)(q,p) = (p/21rli)(W) b (q, p; B) that satisfy the following conditions. For every = b '(W)(q,p)/(21rliZ) densit~operator p on K, every well-defined Hermitian op erator Bon K, every point (q, p) in 9, and every continuous = (1rli) -I tanh(w/3 12) real function R (t) of the real variable t, Xexp(( - 2/w)tanh(w/3 12)H), (46) (i) w(q,p;p) is real and non-negative, (AI) A where the Hamiltonian is (ii) b(q,p;B) is real, (A2) H = jJ2/2m + m{Jif/12. (47) (iii) f f R (w(q,p;p))dq dp = Tr(R (p)), (A3) The quantum mechanical entropy can be obtained by

S(p)=k(lnZ-/3 a~~z) (iv) f f w(q, p; p)R (b (q, p; B))dq dp = Tr( pR (B )). (A4) = (k{3w/2)/tanh(w/3 12) The purpose of seeking this representation is to show - k In(2 sinh(w/3 12)). (4S) that the quantum mechanical entropy is exactly equal to a The Wigner distribution function approaches the classi "classical entropy functional," which is defined by cal canonical distribution function at Ii ~ 0, S(cl)(W) = - k f f w(q,p;p)ln 21T1iw(q,p;p)dqdp. (AS) lim P IW)(q, p) = (m/3 121T)exp( - /3H), (49) 11_0 If such a representation exists, then we choose as predicted by the general considerations. The classical par R (t) = - kIn t and from (AlO) obtain tition function is, by definition, Sip) = - k Tr(p lnp) =s(el)(w). (A6) zlel) = (w/3)-I. (50) Beretta did not know whether this representation exist Finally, the classical entropy has the form ed or not. After making a conjecture, Beretta tried to prove that this representation was just the one to which the leI) k - k In(w/3). (51) S = Wigner, the Blokhintzev, and the Wehrl21 phase-space re With the help ofEqs. (4S) and (51), Eqs. (42) and (43) are presentations (Ro) converge in the classical limit Ii ~ O. maintained. Although we do not know whether this representation exists, we are able to conclude that Ro is an incorrect candi V. CONCLUSIONS date for the representation, the reason being that in Ro, Eqs. We discussed the general phase-space representation of (A3) and (A4) hold only after limit Ii ~ 0 are taken in the quantum mechanics at the classical limit Ii ~ O. We proved right-hand sides. that every representation approaches the same "limit repre Now we tum to consider another problem discussed in sentation" at Ii ~ o. The open question on the relation Ref. 12: the behavior of the spectral expansions in the classi between the classical and quantum mechanical entropies cal limit Ii ~ O. The density operator can be written as fol was answered. The differences between the classical and lows: quantal entropies are shown to approach zero at the classical (A7) limit Ii~O. A where ~ = Ith)( th I is the projector onto the eigenspace ACKNOWLEDGMENTS It/!j) with eigenvalue The author wishes to thank Professor R. F. O'Connell mj = [exp( - f3E ) liZ (AS) for many helpful conversations. j This research was partially supported by the Division of and Materials Science, U. S. Department of Energy under Grant HIt/!) = Ejlt/!)· (AS') No. DE-FG05-S4ER45135. By definition we have

APPENDIX: COMMENTS ON TWO OF THE PROBLEMS (A9) f Pj =1, DISCUSSED IN REF. 12 j=O In a recent paper,12 Beretta gave a set of rather restric where 1 denotes the identity operator. tive conditions defining a complete classical phase space rep- The Wigner equivalences of Eqs. (A 7) and (AS) are

486 J. Math. Phys., Vol. 27, No.2, February 1986 LipoWang 486

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 ao lK. Hepp, Commun. Math. Phys. 35, 265 (1974). p(W)(q,p) = L WiPr)(q,p). (A 10) 21. M. Combes, R. Schrader, and R. Seiler, Ann. Phys. 111, 1 (1978). i=O A 3B. Simon, Commun. Math. Pbys. 71, 247 (1980). The relation between PjW)(q,p) and r(q,p; Pi) in Ref. 12 is 4H. Hogreve, J. Potthoff, and R. Schrader, Commun. Math. Phys. 91, 573 A (1983). PJW)(q,p) = (21rli)-lr(q,p;Pj ). (All) 'E. P. Wigner, Phys. Rev. 40, 749 (1932). Next we consider letting" --. O. It has been shown that22 6K. Husimi, Proc. Phys. Math. Soc. Jpn. 22, 264 (1940). 7K. E. Cahill, and R. J. Glauber, Phys. Rev. 177, 1857, 1882 (1968). 8J. G. Kirkwood, Phys. Rev. 44, 31 (1933). (A12) 9G. S. Agarwal, and E. Wolf, Phys. Rev. D 2, 2161 (1970). 10See the recent reviews, e.g., R. F. O'Connell, Found. Phys. 13, 83 (1983); where Ii is the semiclassical action associated with 1"'i) [i.e., V. I. Tatarskii, Sov. Phys. Usp. 26, 311 (1983); M. Hillery, R. F. O'Con 1= (j + rJli with r the Maslov index]. nell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 123 (1984). While quantization disappears in the classical limit IIA. Wehrl, Rep. Math. Phys.16, 353 (1979). 12G. P. Beretta, J. Math. Phys. 25, 1507 (1984). " --. 0, we expect 13H. J. Groenewold, Physica 12, 405 (1946). 14R. F. O'Connell and Lipo Wang, Phys. Rev. A 30,2187 (1984); 31, 1707 (A13) (1985); Phys. Lett. A 107, 9 (1985). I'Lipo Wang and R. F. O'Connell (to be published). 16N. D. Cartwright, Physica A 83,210 (1976). since Wj is the probability of the system being in state I"'i)' 17 A. J.'E. M. Janssen, J. Math. Phys. 25, 2240 (1984). Thus when li~ -0 is applied to both sides ofEq. (A 10), 18G. C. Summerfield and P. F. Zweifel, J. Math. Phys. 10,233 (1969). the order of1~_o and 1.:=0 cannot be exchanged. Fur 19por recent reviews see, e.g., A. Wehrl, Rev. Mod. Phys. SO, 221 (1978). 2'1(. Imre, E. Ozizmir, M. Rosenbaum, and P. Zweifel, J. Math. Phys. 8, thermore it is easily verified that 1097 (1967). 211t is worth noticing that the Wehrl phase-space representation that Ber (A14) etta referred to is actually the antinormal-ordered (Husimi) one. See, e.g., Ref. 6 and Ref. 7. 22See, e.g., W. H. Miller, Adv. Chem. Phys. 25, 69 (1974); M. V. Berry, in Therefore Eqs. (34), (35), and (39), the conjecture, in Ref. 12 Topics In Nonlinear Dynamics, edited by S. lorna (A. I. P., New York, are not valid. 1978); M. V. Berry, Philos. Trans. R. Soc. London 287, 237 (1977).

487 J. Math. Phys., Vol. 27, No.2, February 1986 LipoWang 487

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