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Journal of Physics A Mathematical and General

Off-diagonal matrix elements in the semiclassical limit

Stephan De Bièvre, Jacques Renaud

Published Dec 7, 1996• 1 citation
Publisher

Abstract

We give a very simple proof of the convergence of the non-diagonal Wigner functions of the harmonic oscillator eigenstates to in the semiclassical limit , and .
DOI
10.1088/0305-4470/29/23/001

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In the context of phase-space quantization, matrix elements and observables result from integration of c-number functions over phase space, with Wigner functions serving as the quasiprobability measure. The complete sets of Wigner functions necessary to expand all phase-space functions include off-diagonal Wigner functions, which may appear technically involved. Nevertheless, it is shown here that suitable generating functions of these complete sets can often be constructed, which are relatively simple, and lead to compact evaluations of matrix elements. New features of such generating functions are detailed and explored for integer-indexed sets, such as for the harmonic oscillator, as well as continuously indexed ones, such as for the linear potential and the Liouville potential. The utility of such generating functions is illustrated in the computation of star functions, spectra, and perturbation theory in phase space.