On phase-space representations of quantum mechanics using Glauber coherent states

Abstract

A phase-space formulation of quantum mechanics is proposed by constructing two representations (identified as pq and qp) in terms of the Glauber coherent states, in which phase-space wave functions (probability amplitudes) play the central role, and position q and momentum p are treated on equal footing. After finding some basic properties of the pq and qp wave functions, the quantum operators in phase-space are represented by differential operators, and the Schrödinger equation is formulated in both pictures. Afterwards, the method is generalized to work with the density operator by converting the quantum Liouville equation into pq and qp equations of motion for two-point functions in phase-space. A coordinate transformation between those points allows one to construct a cell in phase-space, whose central point can be treated as a parameter. In this way, one gets equations of motion describing the evolution of one-point functions in phase-space. Finally, it is shown that some quantities obtained in this paper are related in a natural way with cross-Wigner functions, which are constructed with either the position or the momentum wave functions.

Appendix A. Proof of (75) and (82)

To demonstrate the second equality in (75), consider a pure state described by the momentum wave function $\tilde {\Psi }(p, t)$. The Fourier transform of the μth partial derivative of $\tilde {\Psi }(p, t)$ with respect to p is denoted as Φ _μ (q,t) and given by

$$\begin{array}{@{}rcl@{}} {\Phi}_{\mu}(q, t) &:=& (2\pi \hbar)^{-f/2} \int w(q, p) \tilde{\Psi}^{(\mu)}\left(p, t\right) \mathrm{d}p\\ &=&\left(-\frac{i}{\hbar} q\right)^{\mu} {\Psi}(q, t), \end{array} $$

(A.1)

which is simply related to the position wave function Ψ(q,t). Using the inverse Fourier transform of (A.1), the functions $\tilde {\Psi }^{(\mu )}(p^{\prime } \! -\! \frac {1}{2} p, t)$ and $\tilde {\Psi }^{(\eta )}(p^{\prime } + \frac {1}{2} p, t)$ are written as Fourier transforms of Φ _μ (x,t) and Φ _η (X,t), and inserted in (75), in the definition of ρ _{μ η} (q ^′, p ^′, t). One performs the integration over p by invoking the integral representation of the Dirac delta function, and reorganizes the resulting integral over x by introducing a change of variable of the form $\frac {1}{2}q = q^{\prime } - x$ (see figure 1). Afterwards, the second equality in (A.1) is used, and the resulting expression is reorganized by means of the identity

$$\begin{array}{@{}rcl@{}} {\Psi}^{\star}\!\left(\!q^{\prime}\! -\! \frac{1}{2}q, t\! \right) {\Psi}\! \left(\! q^{\prime} \! +\! \frac{1}{2}q, t \right)\! &=&\! \left\langle \left. q^{\prime} + \frac{1}{2} q \right|{\Psi}(t) \right\rangle\\ &&\times \left\langle {\Psi}(t)\left| q^{\prime} - \frac{1}{2}q\right.\! \right\rangle.\\ \end{array} $$

(A.2)

Finally, by considering all the pure states involved in the density operator $\hat {\rho }(t)$, given by (55), one obtains the second equality in (75).

A similar procedure allows to demonstrate the second equality in (82). Instead of (A.1), one now uses the expression

$$\begin{array}{@{}rcl@{}} \tilde{\Phi}_{\mu}(p, t) &:=& (2\pi \hbar)^{-f/2} \int w^{\star}(q, p) {\Psi}^{(\mu)}\left(q, t\right) \mathrm{d}q\\ &=& \left(+\frac{i}{\hbar} p\right)^{\mu} \tilde{\Psi}(p, t) \end{array} $$

(A.3)

and the identity

$$\begin{array}{@{}rcl@{}} \tilde{\Psi}^{\star}\!\left(p^{\prime} \! -\! \frac{1}{2}p, t \right) \tilde{\Psi}\left(\! p^{\prime}\! +\! \frac{1}{2}p, t \right) \! &=&\! \left\langle \left. p^{\prime}\! +\! \frac{1}{2} p \right| \,{\Psi}(t) \right\rangle\\ &&\times\! \left\langle\! {\Psi}{}({}t{}){} \left|{} p^{\prime}{} -{} \frac{1}{2} p \right. \right\rangle.\\ \end{array} $$

(A.4)