Table of Contents
Research Letters in Physics
Volume 2009, Article ID 298790, 5 pages
Research Letter

Reality of the Wigner Functions and Quantization

1Department of Physics, Zanjan University, P. O. Box 45196-313, Zanjan 45371-38111, Iran
2Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), P. O. Box 45195-1159, Zanjan 45195-1159, Iran

Received 11 January 2009; Accepted 8 May 2009

Academic Editor: Nicholas Bigelow

Copyright © 2009 Sadollah Nasiri and Samira Bahrami. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Here we use the extended phase space formulation of quantum statistical mechanics proposed in an earlier work to define an extended lagrangian for Wigner's functions (WFs). The extended action defined by this lagrangian is a function of ordinary phase space variables. The reality condition of WFs is employed to quantize the extended action. The energy quantization is obtained as a direct consequence of the quantized action. The technique is applied to find the energy states of harmonic oscillator, particle in the box, and hydrogen atom as the illustrative examples.

1. Introduction

Among different methods proposed for the quantum statistical mechanics, one may employ the phase space distribution functions to calculate the average quantities. Wigner [1] pioneered in this direction. Based on the quantum state functions he proposed distribution functions with interesting properties and wide range of applicability. Since then many authors have examined Wigner’s distribution functions, and an amazingly large amount of literature has been amassed. Wigner himself used them to make quantum corrections for thermodynamic equilibrium. Moyal [2] and Husimi [3] developed Wigner’s technique extensively. They also showed that averaging by WFs is equivalent, in quantum mechanics, by an averaging in which noncommutating functions of operators ̂𝑝 and ̂𝑞 are ordered according to Weyl’s ordering rule [4]. Further improvements were made by Margenau and Hill [5], Glauber [6], Mehta [7], O’Connell and Wigner [8], Hillery et al. [9], and Agarwal [10]. Another way of studying the quantum statistical mechanics is proposed by Sobouti and Nasiri [11, 12] by generalizing the principle of least action in configuration space to one in phase space. In this method the phase space coordinates are treated as independent variables on the virtual system point trajectories in phase space. This further permits a definition of extended phase space (EPS), extended lagrangians and hamiltonians. A subsequent canonical quantization procedure or using the path integral technique [13] through these extended entities gives a formulation for quantum statistical mechanics. To demonstrate the simplicity and advantages of the EPS formulation, the technique is applied to solve a few problems. Treatment of Bloch’s theory, partition functions for simple harmonic, and linear potentials are worked out by Nasiri et al. [12]. Obtaining a unique expression for conductivity of a dissipative system of charged particles in different gauges is done using EPS technique by Khademi and Nasiri [14]. Considerable assistance of the EPS technique for studying the dissipative quantum systems is shown by Nasiri and Safari [15] and Razavy [16]. Here we use the later method to obtain an extended hamiltonian governing the evolution of WFs by an appropriate canonical transformation in EPS. An arbitrary extended lagrangian is then obtained by Legender’s transformation. The extended action obtained by this lagrangian is quantized upon the reality condition of WFs that directly yields the quantization of energy. The layout of the paper is as follows. In Section 2, a brief review of the EPS formalism is introduced. In Section 3 the extended hamiltonian and lagrangian are introduced for WFs, and the quantization of extended action and energy are illustrated by a few examples. Section 4 is devoted to conclusions.

2. Background

In this paper we use a direct approach to quantum statistical mechanics that are proposed by extending the conventional phase space and by applying the canonical quantization procedure to the extended quantities in this space (the EPS formalism).

2.1. EPS Formalism

In this formalism an extended lagrangian is introduced as follows [11]:

(𝑝,𝑞,̇𝑝,̇𝑞)=̇𝑝𝑞̇𝑞𝑝+𝐿𝑝(𝑝,̇𝑝)+𝐿𝑞(𝑞,̇𝑞),(1) where𝐿𝑝(𝑝,̇𝑝)and 𝐿𝑞(𝑞,̇𝑞)are lagrangians in p and q space, and 𝑝={𝑝𝑖},𝑞={𝑞𝑖} are the collection of generalized momenta and coordinates specifying a system, respectively. Assuming the phase space coordinates p and q to be independent variables on the virtual trajectories allows one to define momenta 𝜋𝑝 and 𝜋𝑞, conjugate to p and q, as follows:

𝜋𝑝=𝜕=𝜕̇𝑝𝜕𝐿𝑝𝜕̇𝑝𝑞,(2a)𝜋𝑞=𝜕=𝜕̇𝑞𝜕𝐿𝑞𝜕̇𝑞𝑝.(2b) Then, using these extended momenta, one may define an extended hamiltonian as follows:

𝑝,𝑞,𝜋𝑝,𝜋𝑞=𝜋𝑝̇𝑝+𝜋𝑞̇𝑞(𝑝,𝑞,̇𝑝,̇𝑞)=𝐻𝑝+𝜋𝑞,𝑞𝐻𝑝,𝑞+𝜋𝑝.(3) Using the quantization rule [11], one gets the following dynamical equation:

𝑖𝜕𝜒=𝐻𝜕𝜕𝑡=𝜒𝑝𝑖𝜕𝜕𝑞,𝑞𝐻𝑝,𝑞𝑖𝜒=𝜕𝑝𝑛=1(𝑖)𝑛𝜕𝑛!𝑛𝐻𝜕𝑝𝑛𝜕𝑛𝜕𝑞𝑛𝜕𝑛𝐻𝜕𝑞𝑛𝜕𝑛𝜕𝑝𝑛𝜒,(4) where 𝜒(𝑝,𝑞,𝑡)=𝜓(𝑞,𝑡)𝜑(𝑝,𝑡)𝑒(𝑖𝑝𝑞/) is distribution function in EPS where 𝜓(𝑞,𝑡)=(1/2𝜋)𝑑𝑝𝜑(𝑝,𝑡)𝑒𝑖𝑝𝑞/, 𝜋𝑝=𝑖(𝜕/𝜕𝑝), and 𝜋𝑞=𝑖(𝜕/𝜕𝑞). The averaging rule for an observable𝑂(𝑝,𝑞), a c-number operator in this formalism specified by antistandard ordering rule, is

𝑂(𝑝,𝑞)=𝑂(𝑝,𝑞)𝜒(𝑝,𝑞)𝑑𝑝𝑑𝑞.(5) For more details, one may consult Hillery et al. [9] and Sobouti and Nasiri [11].

2.2. The Extended Canonical Transformation and WFs

The quantization procedure of Section 2 assumes p and q as c-numbers, similar to Hamilton’s formulation of classical mechanics. Therefore, the rules of linear canonical transformations in classical mechanics are directly applicable to EPS quantities. These linear canonical transformations correspond to the unitary transformations in quantum mechanics [17].

A simple infinitesimal extended canonical transformation on 𝑝,𝑞,𝜋𝑝,and 𝜋𝑞 gives WFs and the corresponding evolution equation. Let

𝑝𝑝+𝛿𝜆𝜋𝑞,𝜋𝑝𝜋𝑝,(6a)𝑞𝑞+𝛿𝜆𝜋𝑝,𝜋𝑞𝜋𝑞,(6b) where 𝛿𝜆1, the generator of this infinitesimal transformation is

𝐺=𝜋𝑝𝜋𝑞.(7) The corresponding unitary transformation for finite𝜆 is

𝑈=𝑒𝑖𝜆𝐺/=𝑒𝑖𝜆𝜕2/𝜕𝑝𝜕𝑞,𝑈𝑈+=1.(8) The above transformation, for𝜆=1/2, transforms the distribution 𝜒(𝑝,𝑞,𝑡), to those of the Wigner [1]. That is,

=𝜓1𝑊(𝑝,𝑞,𝑡)=𝑈𝜒(𝑝,𝑞,𝑡)𝑞+2𝜓𝜏1𝑞2𝑒𝜏𝑖𝑝𝜏𝑑𝜏,(9) and the evolution equation (4) changes into the Wigner equation:

𝜕𝑊𝜕𝑡=𝑝𝜕𝑊+𝜕𝑞𝑛=01(2𝑛+1)!2𝑖2𝑛𝜕2𝑛+1𝑉𝜕𝑞2𝑛+1𝜕2𝑛+1𝑊𝜕𝑝2𝑛+1.(10) A similar technique could be used to obtain other distribution functions and their corresponding ordering rules [1820].

3. The Extended Hamiltonian and Extended Lagrangian for WFs

By transformation (8) the extended hamiltonian of (3) becomes

𝑤𝑝,𝑞,𝜋𝑝,𝜋𝑞1=𝐻𝑝+2𝜋𝑞1,𝑞2𝜋𝑝1𝐻𝑝2𝜋𝑞1,𝑞+2𝜋𝑝.(11) The subscript 𝑤denotes the quantities for WFs. For 𝐻=𝑝2/2𝑚+𝑉(𝑞) (11) gives

𝑤=112𝑚𝑝+2𝜋𝑞21+𝑉𝑞2𝜋𝑝112𝑚𝑝2𝜋𝑞21𝑉𝑞+2𝜋𝑝.(12) The extended lagrangian, 𝑤, for WFs may be defined by the following Legendre transformation:

𝑤𝑝,𝑞,𝜕𝑤,𝜕̇𝑝𝜕𝑤𝜕̇𝑞=̇𝑝𝜕𝑤𝜕̇𝑝+̇𝑞𝜕𝑤𝜕̇𝑞𝑤(𝑝,𝑞,̇𝑝,̇𝑞).(13) Equation (13) serves as a differential equation to determine 𝑤. Note that the extended lagrangians are not unique and they just serve to obtain the correct equations of motion through the Euler-Lagrange equations for configuration and momentum spaces.

The solutions of the Schrödinger equation can be written as [21]

𝜓(𝑞,𝑡)=𝑅𝑞(𝑞,𝑡)𝑒𝑖𝑆𝑞(𝑞)/,(14) where 𝑆𝑞(𝑞) is the ordinary action in q space defined by

𝑆𝑞(𝐿𝑞)=𝑞(𝑞,̇𝑞,𝑡)𝑑𝑡.(15) By the same manner one may do in p space:

𝜑(𝑝,𝑡)=𝑅𝑝(𝑝,𝑡)𝑒(𝑖𝑆𝑝(𝑝)/),(16) where

𝑆𝑝(𝐿𝑝)=𝑝(𝑝,̇𝑝,𝑡)𝑑𝑡(17) is the action in p space. Substituting for 𝜓 and 𝜑, from (14) and (16) in the expression given for 𝜒, one gets

𝜒=𝜓𝜑𝑒(𝑖𝑝𝑞/)=𝑅𝑞𝑅𝑝𝑒𝑖(𝑆𝑞+𝑆𝑝𝑝𝑞)/.(18) Now, let the extended action be defined as

𝑆𝜒(𝑝,𝑞)=(𝑝,𝑞,̇𝑝,̇𝑞,𝑡)𝑑𝑡.(19) Using (1), (15), and (17) one gets

𝑆𝜒(𝑝,𝑞)=𝑆𝑞+𝑆𝑝𝑝𝑞,(20) which is the same as appeared in the phase of (18). Hence,

𝜒=𝑅𝜒𝑒𝑖𝑆𝜒(𝑝,𝑞)/,(21) where𝑅𝜒(𝑝,𝑞,𝑡)=𝑅𝑝(𝑝,𝑡)𝑅𝑞(𝑞,𝑡). Therefore, in general, any phase space distribution function,𝜌(𝑝,𝑞,𝑡) can be written in the form of an amplitude multiplied by a phase factor [13] as follows:

𝜌(𝑝,𝑞,𝑡)=𝑅𝜌(𝑝,𝑞,𝑡)𝑒𝑖𝑆𝜌(𝑝,𝑞)/,(22) where𝑅𝜌(𝑝,𝑞,𝑡)is the amplitude, and 𝑆𝜌(𝑝,𝑞)is the corresponding extended action of 𝜌(𝑝,𝑞,𝑡). Both of 𝑅𝜌 and 𝑆𝜌 are real functions. In the same manner for WFs, as a possible distribution functions in the phase space, we formally have

𝑊(𝑝,𝑞,𝑡)=𝑅𝑤(𝑝,𝑞,𝑡)𝑒𝑖𝑆𝑤(𝑝,𝑞)/,(23) where

𝑆𝑤(𝑝,𝑞)=𝑤(𝑝,𝑞,̇𝑝,̇𝑞,𝑡)𝑑𝑡,(24) and 𝑤is given by (13). However, it is well known that the WFs are real functions of the phase space variables. Thus, the phase 𝑆𝑤(𝑝,𝑞)in (23) must satisfy the following constraint:

𝑆sin𝑤(𝑝,𝑞)=0,(25) or

𝑆𝑤(𝑝,𝑞)=𝑛𝜋,𝑛=0,1,2,.(26) Equation (26) shows the quantization of extended action. As will be shown later, due to the uncertainty principle, the values of 𝑛must be correctly chosen for a given problem. Note that the extended action in EPS is a function of phase space variables and is basically different from ordinary action 𝑆𝑞(𝑞)defined in configuration space. As an interesting result, one has cos{𝑆𝑤(𝑝,𝑞)/}=cos(𝑛𝜋)=(1)𝑛. Thus, the well known negativity of WFs, 𝑤(𝑝,𝑞,𝑡)=(1)𝑛𝑅𝑤(𝑝,𝑞,𝑡), seems to have a quantum origin [22].

3.1. Energy Quantization

In this section we show that the quantization of the extended action of (26) is directly related to the quantization of energy. To see this we consider a few examples.

3.1.1. Harmonic Oscillator

For harmonic potential, 𝐻=𝑝2/2𝑚+(1/2)𝑚𝜔2𝑞2 (12) gives

𝑤𝑝,𝑞,𝜋𝑝,𝜋𝑞=𝑝𝑚𝜋𝑞𝑚𝜔2𝑞𝜋𝑝.(27) By (13) one gets

𝑝𝑚𝜕𝑤𝜕̇𝑞𝑚𝜔2𝑞𝜕𝑤𝜕̇𝑝=̇𝑝𝜕𝑤𝜕̇𝑝+̇𝑞𝜕𝑤𝜕̇𝑞𝑤(𝑝,𝑞,̇𝑝,̇𝑞).(28) The extended lagrangian as the solution of differential equation (28) may be obtained as

𝑤𝜋(𝑝,𝑞,̇𝑝,̇𝑞)=𝜔𝑝𝑚̇𝑝+𝑚𝜔2𝑞̇𝑞.(29) It can be shown that 𝑤 satisfies the Euler-Lagrange equations in both configuration and momentum space. Using (24) and (29) one obtains

𝑆𝑤𝜋(𝑝,𝑞)=𝜔𝑝2+12𝑚2𝑚𝜔2𝑞2=𝜋𝜔1H2,𝜔(30) where H is the quantum mechanical Hamiltonian assuming the uncertainty principle, implicitly [23]. Equations (26) and (30) give

𝐸𝑛1(𝑝,𝑞)=𝑛+2𝜔,𝑛=0,1,2,3,,(31) as the quantization of energy of a harmonic oscillator.

3.1.2. Particle in the Box

For a particle in a one dimensional box of side L located at the right side of origin, one has

𝑉(𝑞)=,𝑞<0,0,0<𝑞<𝐿,,𝑞>𝐿.(32) For 𝐻=𝑝2/2𝑚 one gets the following from (12):

𝑤𝑝,𝑞,𝜋𝑝,𝜋𝑞=𝑝𝑚𝜋𝑞.(33) By legendre’s transformation on the above extended hamiltonian, one gets for an arbitrary extended lagrangian;

𝑤(𝑝,𝑞,̇𝑝,̇𝑞)=𝐿̇𝑝.(34) Using (24) and (34) the extended action becomes

𝑆𝑤(𝑝,𝑞)=𝐿𝑝=𝐿(2𝑚𝐻)1/2.(35) Then, by (26) and squaring (35) one gets

𝐸𝑛=𝑛2𝜋222𝑚𝐿2,𝑛=1,2,,(36) where 𝑛 cannot assume zero due to the uncertainty principle. Equation (36) shows the energy levels of a particle in the box.

3.1.3. Hydrogen Atom

For the electron in the hydrogen-like atom, as a bound system with Z protons, the hamiltonian is 𝑝2𝑟/2𝑚+𝑙2/2𝑚𝑟2(𝑧𝑒2/𝑟) [24]. By (12) one gets

𝑤𝑝𝑟,𝑟,𝜋𝑝𝑟,𝜋𝑟=𝑝𝑟𝑚𝜋𝑟+22𝑟𝜋𝑝𝑟+2𝑚𝑍𝑒2𝑟2𝜋𝑝𝑟+𝑚𝑧𝑒2𝜋/23𝑝𝑟2𝑚(𝑟(1/2)𝜋𝑝𝑟)2(𝑟+(1/2)𝜋𝑝𝑟)2.(37) A suitable extended lagrangian can be obtained by (25) to be

𝑤𝑝𝑟,𝑟,̇𝑝𝑟=,̇𝑟𝑚𝑍22𝜋𝑒2𝑝𝑟𝑚̇𝑝𝑟2̇𝑟𝑚𝑟3+𝑒2̇𝑟𝑟2×𝑝2𝑟+2𝑚22𝑚𝑟2𝑧𝑒2𝑟3/2.(38) The extended action becomes

𝑆𝑤𝑝𝑟=,𝑟𝑚2𝜋𝑍𝑒2𝐻1/2.(39) Then, by (37) and squaring (39) one gets

||𝐸𝑛||=𝑚𝑍2𝑒422𝑛2=𝑍2𝑒22𝑎0𝑛2,𝑛=1,2,,(40) where 𝑎0 is the Bohr radius, and (40) gives the energy levels of the hydrogen atom. The same as the previous case 𝑛 starts from 1 due to the uncertainty principle.

4. Conclusion

The EPS formulation of quantum statistical mechanics permits the definition of extended hamiltonians and extended lagrangians. A canonical quantization in EPS gives a dynamical equation for the phase space distribution functions. Wigner’s distribution functions and their corresponding evolution equation can be obtained by a simple extended canonical transformation in EPS. The extended hamiltonian through Legendre’s transformation provides a differential equation for the corresponding extended lagrangian. Having the extended lagrangian one may define the extended action as a function of ordinary phase space variables. Applying the reality condition of the WFs yields the quantization of the extended action which in turn gives the energy quantization. The energy levels of the harmonic oscillator, the particle in the box and the hydrogen atom are worked out as a few illustrative examples.


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