Construction of Projection Operators in the Semiclassical Approximation

Abstract

The projection operators onto various subsets of states of a quantum-mechanical system are constructed in a semiclassical approximation based on the Wigner transformation of statistical mechanics. As illustrations, explicit operator expressions are derived for the cases of central Coulomb potential, one-dimensional harmonic oscillator, and radial Coulombic states of specified angular momenta. The accuracy of these operators is then examined in some detail in terms of the overlap integrals and dipole transition probabilities. The semiclassical approximation is found to be effective in the energy regions away from the classical turning points. Extensions of the approach to partially projected Green's functions and other related moments are discussed and their applications to scattering problems pointed out.