Semiclassical approximations in wave mechanics

TLDR

The paper reviews methods for deriving quantum‑mechanical expressions in the semiclassical limit where ℏ is small relative to classical action scales. It discusses WKB connection formulas, complex and comparison‑equation techniques for one‑dimensional and radial problems, semiclassical potential scattering via a three‑stage approximation, and the derivation of Green functions and level densities using Feynman path integrals, highlighting their limitations. All resulting expressions depend on classical orbit families, with analytic forms determined by the topological properties of those families.

Abstract

We review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash is small (in comparison with the relevant classical action functions). To start with we treat one-dimensional problems and discuss the derivation of WKB connection formulae (and their reversibility), reflection coefficients, phase shifts, bound state criteria and resonance formulae, employing first the complex method in which the classical turning points are avoided, and secondly the method of comparison equations with the aid of which uniform approximations are derived, which are valid right through the turningpoint regions. The special problems associated with radial equations are also considered. Next we examine semiclassical potential scattering, both for its own sake and also as an example of the three-stage approximation method which must generally be employed when dealing with eigenfunction expansions under semiclassical conditions, when they converge very slowly. Finally, we discuss the derivation of semiclassical expressions for Green functions and energy level densities in very general cases, employing Feynman's path-integral technique and emphasizing the limitations of the results obtained. Throughout the article we stress the fact that all the expressions obtained involve quantities characterizing the families of orbits in the corresponding purely classical problems, while the analytic forms of the quantal expressions depend on the topological properties of these families.