Wigner operator's new transformation in phase space quantum mechanics and its applications

TL;DR

This paper introduces a new two-fold integral transformation involving the Wigner operator in phase space quantum mechanics, providing a tool for converting between different operator orderings and enriching the phase space framework.

Contribution

The paper derives a novel two-fold integral transformation for the Wigner operator and applies it to relate different operator orderings in phase space quantum mechanics.

Findings

New two-fold integral transformation for Wigner operator derived

Transformation enables conversion between Q-P, P-Q, and Weyl orderings

Enhances the mathematical framework of phase space quantum mechanics

Abstract

Using operators' Weyl ordering expansion formula (Hong-yi Fan,\emph{\}J. Phys. A 25 (1992) 3443) we find new two-fold integration transformation about the Wigner operator $\Delta(q',p')$ ($q$-number transform) in phase space quantum mechanics, \[ \iint_{-\infty}^\infty dp' dq'/\pi \Delta (q',p') e^{-2i(p-p') (q-q')} =\delta (p-P) \delta (q-Q), \] and its inverse \[\iint_{-\infty}^\infty dq dp \delta (p-P) \delta (q-Q) e^{2i(p-p') (q-q')}=\Delta (q',p'), \] where $Q,$ $P$ are the coordinate and momentum operators, respectively. We apply it to studying mutual converting formulas among $Q-P$ ordering, $P-Q$ ordering and Weyl ordering of operators. In this way, the contents of phase space quantum mechanics can be enriched.