On phase-space representations of quantum mechanics using Glauber coherent states
Phase-space quantum mechanics, coherent states, Husimi function, Wigner function
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.