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Physical Review A

Global, uniform semiclassical approximations for quantum systems on the half-line

John R. Klauder

Published Nov 1, 1986• 6 citations
Publisher

Abstract

Motivated by recent experiments on highly excited Rydberg atoms in electrically polarized states, we consider the semiclassical analysis of quantum systems defined on the interval $0<x<\ensuremath{\infty}$. Caustics plague the conventional approach, and Maslov's technique cannot be used. Instead we introduce an affine phase-space formulation that leads to a global, uniform semiclassical approximation to the wave function which fully captures the effects of caustics.

DOI
10.1103/physreva.34.4486

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Motivated by recent experiments on highly excited Rydberg atoms in electrically polarized states, we consider the semiclassical analysis of quantum systems defined on the interval 0<x<\ensuremath∞0<x<\ensuremath{\infty}0<x<\ensuremath∞. Caustics plague the conventional approach, and Maslov's technique cannot be used. Instead we introduce an affine phase-space formulation that leads to a global, uniform semiclassical approximation to the wave function which fully captures the effects of caustics.
Initial-value phase-space integral representations for a time-dependent propagator are obtained in the coordinate and momentum representations. To do so we first derive nonuniform semiclassical propagators for the various representations, obtaining the global-time semiclassical phase indices (Maslov indices) due to caustics. Results include readily implementable general phase index formulas for any type of caustic, including cases where the Morse index theorem is inapplicable. The method of obtaining the indices is general and based simply on concatenating uniform short-time propagators which also gives rise to alternative path-integral forms. Initial-value integral representations are then derived by introducing a method of extending short-time initial-value propagator formulas to global times via a simple stationary-phase asympotoic-equivalence approach. The integrals reduce to the nonuniform semiclassical propagators within the stationary-phase approximation, are uniform about caustics, and have integrand phases which properly account for the global-time phases in terms of appropriate Maslov indices. The initial-value integrals are also consistently derived via a canonical mapping procedure on the coordinate-space path integral. Initial-value integrals for time-dependent wave-function propgation are also given. Evaluation of the initial-value integral expressions do not require trajectory root searches for propagation.