P. Kasperkovitz et M. Peev, SEMICLASSICAL MECHANICS IN ONE-DIMENSION - II - APPROXIMATE MATRIX-ELEMENTS, Journal of physics. A, mathematical and general, 31(9), 1998, pp. 2227-2239
The semiclassical formula for matrix elements, sometimes called the He
isenberg correspondence principle, relates matrix elements (operators
in energy representation) to phase space functions (Weyl representativ
es). The formula does not make sense for arbitrary operators; when it
is valid it implicitly fixes the relative phases of the eigenfunctions
of the Hamiltonian. The conventions, which have to be used for the wa
vefunctions in position or momentum representation, are given here in
explicit form, and we present a class of operators related to coherent
states of high energy for which the Heisenberg correspondence princip
le holds.