Reality of the Wigner Functions and Quantization

Abstract

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 𝑝̂ and 𝑞̂ 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]:

ℒ(𝑝,𝑞,𝑝,̇𝑞̇)=−𝑝𝑞̇−𝑞̇𝑝+𝐿^𝑝(𝑝,𝑝)̇+𝐿^𝑞(𝑞,𝑞̇),(1) where𝐿^𝑝(𝑝,𝑝)̇and 𝐿^𝑞(𝑞,𝑞̇)are lagrangians in p and q space, and 𝑝={𝑝_𝑖},𝑞={𝑞_𝑖} 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 𝜋_𝑝 and 𝜋_𝑞, conjugate to p and q, as follows:

𝜋_𝑝=𝜕ℒ𝜕𝑝̇=𝜕𝜕𝐿𝑝^𝑝̇−𝑞,(2a)𝜋_𝑞=𝜕ℒ𝜕𝑞̇=𝜕𝜕𝐿𝑞^𝑞̇−𝑝.(2b) Then, using these extended momenta, one may define an extended hamiltonian as follows:

ℋ𝑝,𝑞,𝜋_𝑝,𝜋_𝑞==𝜋𝐻_𝑝𝑝̇𝑝++𝜋𝜋_𝑞𝑞_𝑞̇−ℒ(𝑝,𝑞,𝑝,̇𝑞̇),𝑞−𝐻𝑝,𝑞+𝜋_𝑝.(3) Using the quantization rule [11], one gets the following dynamical equation:

𝑖ℏ𝜕𝜒𝜕𝑡==ℋ𝐻𝜒𝑝−𝑖ℏ𝜕=𝜕𝑞,𝑞−𝐻𝑝,𝑞−𝑖ℏ𝜕𝜕𝑝_𝑛=1𝜒(𝑖ℏ)^𝑛𝑛!𝜕^𝑛𝐻𝜕𝑝^𝑛𝜕^𝑛𝜕𝑞^𝑛−𝜕^𝑛𝐻𝜕𝑞^𝑛𝜕^𝑛𝜕𝑝^𝑛𝜒,(4) where 𝜒(𝑝,𝑞,𝑡)=𝜓(𝑞,𝑡)𝜑^∗(𝑝,𝑡)𝑒^{−(𝑖𝑝𝑞/ℏ)} is distribution function in EPS where 𝜓(𝑞,𝑡)=(1/√2𝜋ℏ)∫𝑑𝑝𝜑(𝑝,𝑡)𝑒^{𝑖𝑝𝑞/ℏ}, 𝜋_𝑝=−𝑖ℏ(𝜕/𝜕𝑝), and 𝜋_𝑞=−𝑖ℏ(𝜕/𝜕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,𝜋_𝑝,and 𝜋_𝑞 gives WFs and the corresponding evolution equation. Let

𝑝⟶𝑝+𝛿𝜆𝜋_𝑞,𝜋_𝑝⟶𝜋_𝑝,(6a)𝑞⟶𝑞+𝛿𝜆𝜋_𝑝,𝜋_𝑞⟶𝜋_𝑞,(6b) where 𝛿𝜆≪1, the generator of this infinitesimal transformation is

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

𝑈=𝑒^{𝑖𝜆𝐺/ℏ}=𝑒^{𝑖𝜆ℏ𝜕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,

𝑊(𝑝,𝑞,𝑡)==𝑈𝜓𝜒(𝑞𝑝,+𝑞,1𝑡)2ℏ𝜏𝜓^∗𝑞−12ℏ𝜏𝑒^𝑖𝑝𝜏𝑑𝜏,(9) and the evolution equation (4) changes into the Wigner equation:

𝜕𝑊𝜕𝑡=+−𝑝𝜕𝜕𝑊𝑞_𝑛=01(2𝑛+1)!2ℏ𝑖^2𝑛𝜕^2𝑛+1𝑉𝜕𝑞^2𝑛+1𝜕^2𝑛+1𝑊𝜕𝑝^2𝑛+1.(10) A similar technique could be used to obtain other distribution functions and their corresponding ordering rules [18–20].

3. The Extended Hamiltonian and Extended Lagrangian for WFs

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

ℋ_𝑤=𝑝,𝐻𝑞,𝜋𝑝_𝑝,+𝜋_𝑞12𝜋^𝑞,𝑞−12𝜋^𝑝−𝐻𝑝−12𝜋^𝑞,𝑞+12𝜋^𝑝.(11) The subscript 𝑤denotes the quantities for WFs. For 𝐻=𝑝²/2𝑚+𝑉(𝑞) (11) gives

ℋ_𝑤=12𝑚𝑝+12𝜋^𝑞²+𝑉𝑞−1−12𝜋^𝑝2𝑚𝑝−12𝜋^𝑞²−𝑉𝑞+12𝜋^𝑝.(12) The extended lagrangian, ℒ_𝑤, for WFs may be defined by the following Legendre transformation:

ℋ_𝑤=𝑝𝑝,̇𝑞𝜕,ℒ𝜕_𝑤ℒ𝜕𝑝^𝑤̇,𝜕ℒ𝜕𝑞^𝑤̇𝜕𝑝̇+𝑞̇𝜕ℒ𝜕𝑞^𝑤̇−ℒ^𝑤(𝑝,𝑞,𝑝,̇𝑞̇).(13) Equation (13) serves as a differential equation to determine ℒ_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)=𝑅^𝑞(𝑞,𝑡)𝑒^{𝑖𝑆𝑞(𝑞)/ℏ},(14) where 𝑆^𝑞(𝑞) is the ordinary action in q space defined by

𝑆^𝑞(𝑞)=𝐿^𝑞(𝑞,𝑞̇,𝑡)𝑑𝑡.(15) By the same manner one may do in p space:

𝜑(𝑝,𝑡)=𝑅^𝑝(𝑝,𝑡)𝑒^{−(𝑖𝑆𝑝(𝑝)/ℏ)},(16) where

𝑆^𝑝(𝑝)=𝐿^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=𝜓𝜑^∗𝑒^{−(𝑖𝑝𝑞/ℏ)}=𝑅^𝑞𝑅^𝑝𝑒^{𝑖(𝑆𝑞+𝑆𝑝−𝑝𝑞)/ℏ}.(18) Now, let the extended action be defined as

𝑆_𝜒(𝑝,𝑞)=ℒ(𝑝,𝑞,𝑝,̇𝑞̇,𝑡)𝑑𝑡.(19) Using (1), (15), and (17) one gets

𝑆_𝜒(𝑝,𝑞)=𝑆^𝑞+𝑆^𝑝−𝑝𝑞,(20) which is the same as appeared in the phase of (18). Hence,

𝜒=𝑅_𝜒𝑒^{𝑖𝑆𝜒(𝑝,𝑞)/ℏ},(21) where𝑅_𝜒(𝑝,𝑞,𝑡)=𝑅^𝑝(𝑝,𝑡)𝑅^𝑞(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)=𝑅_𝜌(𝑝,𝑞,𝑡)𝑒^{𝑖𝑆𝜌(𝑝,𝑞)/ℏ},(22) where𝑅_𝜌(𝑝,𝑞,𝑡)is the amplitude, and 𝑆_𝜌(𝑝,𝑞)is the corresponding extended action of 𝜌(𝑝,𝑞,𝑡). Both of 𝑅_𝜌 and 𝑆_𝜌 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)=𝑅_𝑤(𝑝,𝑞,𝑡)𝑒^{𝑖𝑆𝑤(𝑝,𝑞)/ℏ},(23) where

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

sin𝑆^𝑤(𝑝,𝑞)ℏ=0,(25) or

𝑆_𝑤(𝑝,𝑞)=𝑛𝜋ℏ,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 𝑆^𝑞(𝑞)defined in configuration space. As an interesting result, one has cos{𝑆_𝑤(𝑝,𝑞)/ℏ}=cos(𝑛𝜋)=(−1)^𝑛. Thus, the well known negativity of WFs, 𝑤(𝑝,𝑞,𝑡)=(−1)^𝑛𝑅_𝑤(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

ℋ_𝑤𝑝,𝑞,𝜋_𝑝,𝜋_𝑞=𝑝𝑚𝜋^𝑞−𝑚𝜔²𝑞𝜋_𝑝.(27) By (13) one gets

𝑝𝑚𝜕ℒ𝜕𝑞^𝑤̇−𝑚𝜔²𝑞𝜕ℒ^𝑤𝜕𝑝̇=𝑝̇𝜕ℒ𝜕𝑝^𝑤̇+𝑞̇𝜕ℒ𝜕𝑞^𝑤̇−ℒ^𝑤(𝑝,𝑞,𝑝,̇𝑞̇).(28) The extended lagrangian as the solution of differential equation (28) may be obtained as

ℒ_𝑤(𝑝,𝑞,𝑝,̇𝑞̇)=𝜋𝜔𝑚𝑝𝑝̇+𝑚𝜔²𝑞𝑞̇.(29) It can be shown that ℒ_𝑤 satisfies the Euler-Lagrange equations in both configuration and momentum space. Using (24) and (29) one obtains

𝑆_𝑤(𝑝,𝑞)=𝜋𝜔2𝑝𝑚²+12𝑚𝜔²𝑞²=𝜋𝜔H−12ℏ𝜔,(30) where H is the quantum mechanical Hamiltonian assuming the uncertainty principle, implicitly [23]. Equations (26) and (30) give

𝐸_𝑛(𝑝,𝑞)=𝑛+12ℏ𝜔,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)=⎧⎪⎨⎪⎩∞0∞,,,𝑞0𝑞<><0𝐿𝑞,.<𝐿,(32) For 𝐻=𝑝²/2𝑚 one gets the following from (12):

ℋ_𝑤𝑝,𝑞,𝜋_𝑝,𝜋_𝑞=𝑝𝑚𝜋^𝑞.(33) By legendre’s transformation on the above extended hamiltonian, one gets for an arbitrary extended lagrangian;

ℒ_𝑤(𝑝,𝑞,𝑝,̇𝑞̇)=𝐿𝑝.̇(34) Using (24) and (34) the extended action becomes

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

𝐸_𝑛=𝑛²𝜋²ℏ²2𝑚𝐿²,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 𝑝^2𝑟/2𝑚+𝑙²/2𝑚𝑟²−(𝑧𝑒²/𝑟) [24]. By (12) one gets

ℋ_𝑤=𝑝_𝑟𝑝,_𝑟𝑟,𝜋_𝑝𝑟,𝜋_𝑟𝑚𝜋^𝑟+2ℓ²𝑟𝜋_𝑝𝑟+2𝑚𝑍𝑒²𝑟²𝜋_𝑝𝑟+𝑚𝑧𝑒²/2𝜋^3𝑝_𝑟2𝑚(𝑟−(1/2)𝜋_𝑝𝑟)²(𝑟+(1/2)𝜋_𝑝𝑟)².(37) A suitable extended lagrangian can be obtained by (25) to be

ℒ_𝑤𝑝_𝑟,𝑟,𝑝_𝑟̇,𝑟̇=𝑚𝑍2√2𝜋𝑒²𝑝𝑚^𝑟𝑝̇^𝑟−ℓ²𝑟̇𝑚𝑟³+𝑒²𝑟̇×𝑝_2𝑟𝑟²2𝑚+2𝑚ℓ²𝑟²−𝑧𝑒²𝑟^−3/2.(38) The extended action becomes

𝑆_𝑤𝑝_𝑟,𝑟=𝑚2𝜋𝑍𝑒²𝐻^−1/2.(39) Then, by (37) and squaring (39) one gets

||𝐸_𝑛||=𝑚𝑍²𝑒⁴2ℏ²𝑛²=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.