A quantum mechanical representation in phase space

Abstract

A quantum mechanical representation suitable for studying the time evolution of quantum densities in phase space is proposed and examined in detail. This representation on 2'2 (2) phase space is based on definitions of the operators P and Q in phase space that satisfy various correspondences for the Liouville equation in classical and quantum phase space, as well as quantum position and momentum 2'2 (1) spaces. The definitions presented here, P=p/2-ifti)/aq and Q=q/2+ifza/ap, are related to definitions that have been recently proposed [J. Chern. Phys. 93, 8862 (1990)]. The resulting quantum phase space representation shares many of the mathematical properties of usual representations in coordinate and momentum spaces. Within this representation, time evolution equations for complex-valued functions (wave functions) and their square magnitudes (distribution functions) are derived, and it is shown that the coordinate and momentum space time evolution equations can be recovered by a simple Fourier projection. The phase space quantum probability conservation equation obtained is a good illustration of the quantization rule that requires one to replace the classical Poisson bracket between the Hamiltonian and the probability density with the quantum commutator between the corresponding operators. The possible classical analogs to quantum probabilities densities are also considered and some of the present results are illustrated for the dynamics of the coherent state. Tr(pp') =0 and then considering the integrand in Eq.

Wavefunctions on phase space

“…A main goal for this paper was to show that the quantum structure envisioned by Torres-Vega and co-workers [5] and by Harriman [6], equations (1) to (4), can be realized by 'lifting' the usual formalism with coherent states up a dimension. Then over-completeness (equation (10)) becomes simple completeness in the augmented space (equations (47) and (48)).…”

“…Equations (40), (41), (47), (48) and (50) provide an expression of the full proposal of Torres-Vega et al [5] for the cases α = 1 = γ, β = 1/2 = −δ and…”

“…To try to achieve this Go. Torres-Vega and co-workers [5] and, independently, Harriman [6] have hypothesized a phase space representation which they write as | Γ = |p, q corresponding to 'eigenstates' of some (unspecified) operatorΓ so that, Γ|Γ = Γ |Γ and Γ | Γ = δ(Γ − Γ ) .…”

“…The formalism in equations (1) to (4) was suggested ( [5], [8]) by the properties of coherent states. In their generalized form these are [9] |p, q; σ ≡D [p, q] |σ ,…”

Quantum Potential and Symmetries in Extended Phase Space

“…Although, the arguments invoked on the basis of the Heisenberg uncertainty principle made the physical meaning of the "phase space points" problematic, things have changed and phase space techniques mainly formulated by the theory of deformation quantization [23] and a family of Schrödinger equations in phase space [24,25,26,27,28] are now widely accepted and used. As an approach of the latter type, Sobouti and Nasiri [29] have proposed a formulation of quantum statistical mechanics by generalizing the principle of least action to the trajectories in phase space, and a canonical quantization procedure in this space.…”