E. P. Wigner. On the quantum correction for thermodynamic equilibrium. Phys. Rev., 40:749{ 759, 1932.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and allow us to prove in a fairly simple and straightforward manner self averaging of the time reversed signal. We recall that the Wigner transform is a convenient tool to analyze high frequency wave propagation in deterministic [18, 20, 27] and random media [24] Introduced by Wigner in 1932 [28], it has been used extensively in the mathematical literature recently. Convergence of the average Wigner distribution to the solution of the radiative transfer equation was rst proved by H. Spohn in [25] for time independent potentials on small time intervals. This result was extended to global ....

E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40, 1932, 749-759. 21

....and allow us to prove in a fairly simple and straightforward manner self averaging of the time reversed signal. We recall that the Wigner transform is a convenient tool to analyze high frequency wave propagation in deterministic [19, 21, 28] and random media [25] Introduced by Wigner in 1932 [29], it has been used extensively in the mathematical literature recently. Convergence of the average Wigner distribution to the solution of the radiative transfer equation was rst proved by H. Spohn in [26] for time independent potentials on small time intervals. This result was extended to global ....

E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40, 1932, 749-759. 21

....#) be the instantaneous auto correlation of a complex signal g(t) defined as R g (t, #) g(t # 2)g # (t # 2) 4.1) where g # denotes the complex conjugate of g. The Wigner distribution of g(t) is then defined as the Fourier transform (FT) of R g (t, #) with respect to the lag variable # [151]: R g (t, #)e j## d# = g(t # 2)g # (t # 2)e j## d# , 4.2) or equivalently as W g (t, #) 1 G(# # 2)G # (# # 2)e d# , 4.3) where G(#) is the Fourier transform of g(t) the range of integrals is from to # unless otherwise stated) Similarly, but with a di#erent ....

E. Wigner, "On the quantum correction for thermodynamic equilibrium", Phys. Rev., Vol. 40, 1932, pp. 749--759.

....dynamical behavior of a classical (quasi )distribution function at constant energy in phase space. The CLE can be discretized the by particle methods with di#erent particle shape functions. A common approach is to approximate the phase space distributions by collections of Dirac functionals (cf. [17]) In this case, the dynamics is reduced to Newton s equations of motion, which are routinely solved in classical molecular dynamics simulations. The attractive simplicity of such a local particle base has, however, two major drawbacks. First of all, Dirac functions representation is hardly ....

E. P. Wigner. On the quantum correction for thermodynamic equilibrium. Phys. Rev., 40:749--759, 1932. 16

....University Villanova, Pa 19085 The fiftieth anniversary of the IEEE Signal Processing Society also marks fifty years since Ville [447] applied Wigner distribution (WD) to signal analysis. The Wigner distribution (WD) which was the first distribution introduced in the context of quantum mechanics [452], has paved the way to several key contributions to advances in the area of time frequency distributions as well as representations of signals with time varying characteristics. These contributions have aimed at overcoming the drawbacks of the WD and sought new, more effective tools for ....

E. P. Wigner, "On the quantum correction for thermodynamic equilibrium," Physics Review, vol. 40, pp. 749--759, 1932.

....by Fourier s law q = GammarT , where T is the electron temperature. Quantum mechanical effects appear in the stress tensor and the energy density. Ref. 2] derives the stress tensor and the energy density for the O(h 2 ) momentum shifted thermal equilibrium Wigner distribution function [8]: P ij = GammanT ffi ij h 2 n 12m 2 x i x j log(n) O(h 4 ) 5) 3 W = 3 2 nT 1 2 mnu 2 Gamma h 2 n 24m r 2 log(n) O(h 4 ) 6) Ancona, Iafrate, and Tiersten [9, 10] derived expression (5) for the stress tensor. In Ref. 1] Grubin and Kreskovsky formulated a ....

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Physical Review, vol. 40, pp. 749--759, 1932.

....of suitable observables converges to the Poisson bracket of the limits. For a large class of convergent Hamiltonians the h wise action of the corresponding dynamics converges to the classical Hamiltonian dynamics. The connections with earlier approaches, based on the WKB method, or on Wigner distribution functions, or on the limits of coherent states are reviewed. Physics and Astronomy classification scheme PACS (1994) 03.65.Sq, 03.65.Db 1 FB Physik, Universitat Osnabruck, 49069 Osnabruck, Germany 2 Electronic mail: reinwer dosuni1.rz.Uni Osnabrueck.DE 1 1. Introduction ....

....in WKB terms. The WKB wave functions do correspond to (a subclass of ) convergent states in our approach (see Section 4.8) Their limits are measures supported by Lagrangian manifolds in phase space, hence they have a curious intermediate position between point measures and general measures. B) Wigner functions. Wig,BB,Bru,BCSS,Ara] It is often claimed that quantum mechanics has an equivalent reformulation in terms of Wigner s phase space distribution functions. The classical limit could then be stated very simply in terms of these functions. However, the premise is only partly correct. ....

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E.P. Wigner: "On the quantum correction for thermodynamic equilibrium", Phys.Rev. 40(1932) 749--759

....describe. The Wigner transform of the density matrix is the Fourier transform of the function ae i x 2 j; x Gamma 2 j; t j with respect to j, i.e. w (x; v; t) 1 (2) d Z IR d ae i x 2 j; x Gamma 2 j; t j e iv Deltaj dj (3. 1) cf. 5] 8] [17]) where the Fourier transform is defined by (v) 1 (2) d Z IR d x (x)e iv Deltax dx: 3.2) It is the solution of the Wigner Hartree Fock equation, obtained from (1.1) by an easy calculation [11] w t v Delta r x w [V E ]w [V H ]w Omega [w ....

E. Wigner, On the Quantum Correction for the Thermodynamic Equilibrium, Phys. Rev. 40, pp. 749-759, 1932. 9

....x y=2 and x Gamma y=2 in (13) allow to read the oscillations of u on this scale, and to get much better informations than the mere weak convergence of u in some weighted L 2 space. These two remarks briefly motivate the introduction of f as in (13) but we refer to [11], 7] 6] among others for a more complete discussion of the Wigner transform. We now turn to computing the equation satisfied by f . Using the notation, b f (x; y) F Gamma1 y f (x; u (x y 2 ) u (x Gamma y 2 ) it is classical to observe the identity, ....

E. Wigner. On the quantum correction for thermodynamic equilibrium. Phys. Rev., 40, 1932.

....distribution of u(t) Because the signal now enters twice into the representation the representation is called a quadratic time frequency representation. In Appendix A we discuss some of the mathematical properties of the Wigner distribution. Originally, the Wigner distribution was derived by Wigner (1932) for the calculation of the quantum correction terms to the Boltzmann formula. In the early eighties Claasen and Mecklenbrauker developed a comprehensive approach for the application of the Wigner distribution to joint time frequency analysis (Claasen and Mecklenbrauker, 1980) Figure 2 shows the ....

Wigner, E.P., 1932, On the quantum correction for thermo-dynamic equilibrium: Physical Review, 40, 749-759.

....relaxation terms can be derived. The above expressions are obtained by assuming that the colli2 sion operator in the Wigner Boltzmann equation is such that the corresponding moments equal C J and CE , respectively [13] The quantum correction to the energy density was rst derived by Wigner [38]. The hydrodynamic formulation of quantum systems has been used already by Madelung in 1927 [23] In special situations, simpler hydrodynamic models can be (formally) derived. For instance, assuming that the temperature only depends on the particle density via T = T (n) n with 1, one ....

E. Wigner. On the quantum correction for thermodynamic equilibrium. Phys. Rev., 40:749-759, 1932.

....2 3 2 kB T 2 24m log(n) the term Q = 2 2qm p n p n is the Bohm quantum correction, and CE is the energy relaxation term de ned in (9) The heat ux rT comes from the closure condition. The quantum correction to the energy density was rst derived by Wigner [219]. The hydrodynamic formulation of quantum systems has been used already by Madelung in 1927 [156] Eqs. 32) 35) are considered in the whole space R 3 or in a bounded domain with initial conditions for n, J n and E. The question of appropriate boundary conditions is delicate; see [87, 134, ....

E. Wigner. On the quantum correction for thermodynamic equilibrium. Phys. Rev., 40:749-759, 1932.

....potential is V . The electron density n is defined by n (t, x) j#N l j y j 2 , where the l j 0 are the mixed state occupation probabilities. They verify j#N l j = 1. The transition from the quantum model to the classical one has been done first by Wigner [13], who introduced the Wigner transform of the wave function y. This kind of limit has been performed rigorously by P. Gerard [14] and P. L. Lions and T. Paul [6] N. Mauser and P. Markowich [15] in the linear case and for Schrodinger Poisson by using Wigner measure. We can do exactly the same in ....

Wigner E., On the quantum correction for the Thermodynamic equilibrium, Phys. Rev. 40 (1932), 749-759. 22

....time frequency representation given by WV [f ] x, #) 1 2# Z IR f(x t 2)f(x t 2)e it# dt, 2.1) for all f # L 2 (IR) In the sequel we will refer to the domain of the Wigner distribution as the Wigner plane. This representation was already introduced in 1932 by Wigner in his paper [43]. He presented this representation in the field of quantum mechanics. In 1948, Ville introduced the representation in the fields of signal analysis in [39] Therefore, this representation is also known in the literature as the Wigner Ville distribution. Later in this report we will also use the ....

E.P. Wigner, "On the quantum correction for thermodynamic equilibrium", Phys. Review, 40, 749759, 1932.

....Thus the high frequency limit for the focusing NLS, without any regularizing mechanisms, does not make sense physically or mathematically. 3 The Wigner distribution An essential step in deriving phase space transport equations from wave equations is the introduction of the Wigner distribution [16, 12]. We begin with a brief review of some basic facts and then give the phase space form of the high frequency limit. For any smooth function , rapidly decaying at in nity, the Wigner distribution is de ned by W (x; k) 1 2 3 Z R 3 e ik y (x y 2 ) x y 2 )dy (3.1) where is ....

E.Wigner, On the quantum correction for thermodynamic equilibrium, Physical Rev., 40, 1932, 749-759.

....Wigner spectrum is a convergent sequence of discrete points. PACS (1993) Classification: 03.65.Db, 05.30.Ch, 02.30.Px, 42.50.Dv 1 FB Physik, Universitat Osnabruck, 49069 Osnabruck, Germany 2 Electronic mail: reinwer dosuni1.rz.Uni Osnabrueck.DE 1. Introduction Wigner s distribution function [1,2] associates with each state of a finite non relativistic quantum system a distribution function on phase space. At first sight this seems to contradict Heisenberg s uncertainty principle, since such a distribution function provides a joint probability density for position and momentum. There is no ....

.... That this transform really deserves the name Fourier transform, and has many properties analogous to the classical Fourier transform (with the symplectic form used as a the scalar product between the two sets of variables) is shown in detail in [8] Wigner s phase space distribution function [1] is defined by taking this analogy literally, i.e. by interpreting F ae as the Fourier transform of an ordinary function, then called the Wigner function W ae of ae. This way of computing the Wigner function is a bit cumbersome, however, and it is often better to use the equivalent formula [13] ....

E.P. Wigner:"On the quantum correction for thermodynamic equilibrium", Phys.Rev. 40(1932) 749-759

....to be harmonizable if Z 1 Gamma1 Z 1 Gamma1 j Phi( 1 ; 2 )jd 1 d 2 1: 6.14) In this case, fl(s; t) Z 1 Gamma1 Z 1 Gamma1 e i( 1 s Gamma 2 t) Phi( 1 ; 2 )d 1 d 2 ; 6.15) 6.3. LOCALLY STATIONARY PROCESSES 55 which is the analogous of (6. 6) For deterministic signals, Wigner(1932) introduced in quantum mechanics the function W (t; Z 1 Gamma1 x(t =2)x(t Gamma =2)e Gammai d ; 6.16) called the Wigner Ville distribution, due to the fact that Ville(1948) also used it in signal theory. In the case of a continuous stochastic process, the Wigner Ville spectrum ....

Wigner, E.P.(1932). On the quantum correction for thermodynamic equilibrium.

....solution can be constructed by the method of characteristics and singularities form when these characteristics (rays) cross. 2. 2 The Wigner distribution An essential step in our approach to deriving radiative transport equations from wave equations is the introduction of the Wigner distribution [41]. For any smooth function OE, rapidly decaying at infinity, the Wigner distribution is defined by W (x; k) 1 2 d Z R d e ik Deltay OE(x Gamma y 2 )OE(x y 2 )dy (2.11) where OE is the complex conjugate of OE and the dimension d = 2 or 3. The Wigner distribution is defined ....

E.Wigner, On the quantum correction for thermodynamic equilibrium, Physical Rev., 40, 1932, 749-759.

.... rr 0 q n q m q r : 4 Densities For one single variable, or in the commutative case, one can use Bochner s theorem to associate a density to a quantum random variable, cf. 14] We now want to associate joint densities to several non commutating variables, along the line of Wigner distributions [17]. We will map functionals on an algebra with n generators to measures on IR n . Equivalently, we can ask for a map from functions on IR n (e.g. polynomials) to elements of the algebra. Consider the following diagram: Wigner QS Gamma CS Duality l l Duality QO Gamma CO Weyl where QS: ....

E. Wigner. On the quantum corrections for thermodynamic equilibrium, Phys. Rev. 40, 749-759, 1932.

....distribution: W x (t; 2e jt =2 STFT x (2t; 2 ; h) where h( x ( The Wigner distribution is more commonly presented in the following form: W x (t; Z x(t 2 ) x (t Gamma 2 ) e Gammaj d (2. 2) and was derived in the context of quantum mechanics by Wigner [73] and later extended to the signal processing context by Ville [72] The Wigner distribution satisfies many of the desirable properties in Table 2.1 (all except positivity, strong time support, and strong frequency support) Perhaps the most remarkable property of the Wigner distribution is that ....

E. P. Wigner, "On the quantum correction for thermodynamic equilibrium," Physical Review, vol. 40, pp. 749--759, 1932.

....otherwise impossible for such an abelian group. Summing up, as a realization of the classical quantum correspondence principle, Weyl proposed with (4) to consider a passage from the usual scalar Fourier transform to an operatorial one written in terms of projective representations. The inverse way [29], leading from the quantum to the classical picture, involves an integration on operator space (taking of the trace) f(q; p) Tr[S y (q; p) f ] 6) Since the projective operator product (5) carries a natural twisting, and since formulas (4) 6) ought to represent an algebra ....

Wigner, E.P. (1932). On the Quantum Correction for Thermodynamic Equilibrium, Phys. Rev. 40, 749-759.

....Germany. 2 Electronic mail: reinwer dosuni1.rz.Uni Osnabrueck.DE The study of non relativistic quantum mechanics in terms of an associated classical phase space is an old subject. Perhaps the most widely known map taking quantum states to functions on phase space is the Wigner transformation [19]. However, since the Wigner function of a state need not be integrable, it often represents a probability density, in which an infinite positive probability is cancelled by an infinite negative probability to give formally the normalization to unity. These infinities make the Wigner function ....

E.P. Wigner:"On the Quantum Correction for Thermodynamic Equilibrium", Phys.Rev. 40 (1932) 749

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E. P. Wigner. On the quantum correction for thermodynamic equilibrium. Phys. Rev., 40:749{ 759, 1932.

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E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev., vol. 40, pp. 749--759, 1932.

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Wigner, E.P. On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, (1934), 742-759. 41

....[27] In contrast, the energy distributions, another class of time frequency representations, distribute the energy of the signal over two description variables: time and frequency. In order to devise a joint function of time and frequency, a lot of distributions have been proposed since 1932 [28 32] but only in 1966 Cohen [33] proved that an infinite number of distributions can be generated by the unified formulation: r z t; f j2py u2t gy ; tzu zu2 2j2pf t dy du dt where g(y ,t ) is an arbitrary function called the Kernel function. The Kernel function characterizes the ....

E.P. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932) 749 -- 759.

....Let R (t,#) be the instantaneous auto correlation of a complex signal g(t) defined as (t,#) g(t## 2)gH(t # 2) 1) where gH denotes the complex conjugate of g. The Wigner distribution of g(t) is then defined as the Fourier transform (FT) of R (t,#) with respect to the lag variable # [36], t,#)e#### d# g(t## 2)gH(t # 2)e#### d#, 2) or equivalently as 1 G(### 2)GH(# # 2)e### d#, 3) where G(#) is the Fourier transform of g(t) the range of integrals is from R to #R unless otherwise stated) The WD satisfies a large number of desirable mathematical properties ....

E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932) 749---759.

....This is the case, mainly because the necessary higher order corrections to the adiabatic limit are dicult to achieve, at least by means of the typical techniques like WKB expansions or matched asymptotics. Herein, it is proposed to gain higher order expansions by means of the Wigner transform [57] and its asymptotic properties as studied by G erard, Markowich et al. in their remarkable article [20] This will permit us to analyze the non adiabatic behavior of the QCL: we will not only present a rst order approximation result for di erent variants of the QCL (which can be generalized ....

....result for di erent variants of the QCL (which can be generalized to higher orders, see Sec. 4) but will also demonstrate that the QCL allows to approximate non adiabatic quantum e ects near avoided crossings (Sec. 6) In this approach, the QCL will result from a partial Wigner transform [57] of the full Schr odinger equation of the system. The present author expects that the primary importance of the QCL lies in the fact that it allows the construction of quantum classical particle methods, either of deterministic character as in [16] or stochastic ones leading to surfacehopping ....

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E. Wigner. On the quantum correction for thermodynamic equilibrium. Phys. Rev., 40:749-759, 1932.

....This is the case, mainly because the necessary higher order corrections to the adiabatic limit are dicult to achieve, at least by means of the typical techniques like WKB expansions or matched asymptotics. Herein, it is proposed to gain higher order expansions by means of the Wigner transform [57] and its asymptotic properties as studied by G erard, Markowich et al. in their remarkable article [20] This will permit us to analyze the non adiabatic behavior of the QCL: we will not only present a rst order approximation result for di erent variants of the QCL (which can be generalized to ....

....result for di erent variants of the QCL (which can be generalized to higher orders, see Sec. 4) but will also demonstrate that the QCL allows to approximate non adiabatic quantum e ects near avoided crossings (Sec. 6) In this approach, the QCL will result from a partial Wigner transform [57] of the full Schr odinger equation of the system. The present author expects that the primary importance of the QCL lies in the fact that it allows the construction of quantum classical particle methods, either of deterministic character as in [16] or stochastic ones leading to surfacehopping ....

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E. Wigner. On the quantum correction for thermodynamic equilibrium. Phys. Rev., 40:749-759, 1932.

....g W, iii) medium high temperature: W k B T= We now introduce the Wigner function W of the electron ensemble, defined on the phase space R d x R d x for t 0: W (x;x; t) 1 (2p) d Z R d h r(x 2m h;x 2m h; t)e ix h dh 2 L 2 (R d x R d x ) 2. 5) cf. [37], 27] 28] 22] 26] 29] Note that the self adjointness of R(t) implies r(x;y; t) r(y;x; t) stands for complex conjugation) which in turn implies that W is real valued. It is a simple exercise to compute the evolution equation satisfied by (2. 5) From (2. 2) we obtain the ....

Wigner, E. On the quantum correction for thermodynamic equilibrium. Phys. Rev., 40:749--759, 1932. 38

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E.P. Wigner (1932): On the quantum corrections for the thermodynamic equilibrium. Phys. Rev.40, 749759

....electron ensemble in quantum semiconductor devices (see [30, 16] and it reads ( 1, 6, 13] w t (v r x )w e m T h=m [ w = 1 LQFP w; 1.7) coupled to the Poisson equation (1. 2) Here, w(t; x; v) is the (real valued, but not pointwise non negative) Wigner distribution function (see [32, 20] for a detailed discussion of its properties) As in the above classical case, the charge and current densities are respectively given by (w) t; x) Z IR N w(t; x; v)dv; j(w) t; x) Z IR N vw(t; x; v)dv; x 2 ; t 0: 1.8) 3 While w(t; x; v) may take negative values, w) t; x) ....

E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40, 749-759, 1932. 28

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E. Wigner, "On the Quantum Correction for Thermodynamic Equilibrium," Physics Review,

....in quantum semiconductor devices (see [30, 16] and it reads ( 1, 6, 13] w t (v Delta r x )w Gamma e m T h=m [ Phi]w = 1 LQFP w; 1.7) coupled to the Poisson equation (1. 2) Here, w(t; x; v) is the (real valued, but not pointwise non negative) Wigner distribution function (see [32, 20] for a detailed discussion of its properties) As in the above classical case, the charge and current densities are respectively given by ae(w) t; x) Z IR N w(t; x; v)dv; j(w) t; x) Z IR N vw(t; x; v)dv; x 2 Omega ; t 0: 1.8) While w(t; x; v) may take negative values, ae(w) t; ....

E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40, 749--759, 1932.

....be the instantaneous auto correlation of a complex signal g(t) defined as R g (t; g(t =2)g (t Gamma =2) 1) where g denotes the complex conjugate of g. The Wigner distribution of g(t) is then defined as the Fourier transform (FT) of R g (t; with respect to the lag variable [36]: W g (t; Z R g (t; e Gammaj d = Z g(t =2)g (t Gamma =2)e Gammaj d ; 2) or equivalently as W g (t; 1 2 Z G( 2)G ( Gamma =2)e j t d ; 3) where G( is the Fourier transform of g(t) the range of integrals is from Gamma1 to 1 unless ....

E. Wigner, "On the quantum correction for thermodynamic equilibrium", Phys. Rev., Vol. 40, 1932, pp. 749--759.

....aperiodic or periodic. 1 The four types of signals are listed in Table 1 along with the signal properties in the time domain and the corresponding Fourier transform. The Wigner (or Wigner Ville) distribution was originally defined for type I signals and is usually presented in the following form [1, 2, 3, 4, 5]: W I x (t; Z x(t 2 ) x (t Gamma 2 ) e Gammaj d (1) Since there are four types of Fourier transforms it is reasonable to assume that there could potentially be four types of Wigner distributions. To avoid confusion, the original Wigner distribution will be referred ....

....plane [6, 7] but this is not necessarily the same as computing a type II Wigner distribution. For comparison, note that the type II spectrogram is not a sampled version of the type I spectrogram [8] II Definitions of the Wigner Distribution The Wigner distribution was originally defined [1, 2] as in equation 1 with no obvious means for extending the definition to the three types of discrete signals presented above. We will briefly present three methods that have been used to create discrete Wigner distributions and indicate their shortcomings. We will then present a fourth method that ....

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E.P. Wigner. On the quantum correction for thermodynamic equilibrium. Physical Review, 40:749--759, 1932.

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E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932), 749-759.

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E.P. Wigner. On the quantum correction for thermodynamic equilibrium. Physics Review, 40:749-- 59, 1932.

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E. P. Wigner, "On the Quantum Correction for Thermodynamic Equilibrium," Physical Review, 40, 749-759 (1932).

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E.P. Wigner. On the quantum correction for thermodynamic equilibrium. Physics Review, 40:749-- 59, 1932. 27

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E.P. Wigner. On the quantum correction for thermodynamic equilibrium. Physics Review, 40:749--59, 1932.

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]E.P. Wigner, On the Quantum correction for thermodynamic equilibrium, Phys. Rev. (1932), 749--759.

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E. P. Wigner, \On the quantum correction for thermodynamic equilibrium," Physical Review 40, pp. 246-254, 1932.

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E.P. Wigner (1932): On the Quantum Corrections for the Thermodynamic Equi librium. Phys. Rev.40, 749-759

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E.P. Wigner (1932): On the quantum corrections for the thermodynamic equilibrium. Phys. Rev.40, 749759

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E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40, 749--759 (1932).

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E. Wigner (1932). On the quantum corrections for thermodynamic equilibrium, Phys. Rev. 40, 749-759.

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E.Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40, 1932, 749-759.

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E. Wigner, On the quantum correction for thermodynamic equilibrium, Physical Rev., 40, 1932, 749-759.

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Wigner, E.: On the Quantum Correction for Thermodynamic Equilibrium. Phys. Review, vol. 40, June 1, 1932, pp. 749--759.

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