Semiclassical approximations in wave mechanics

1972/01/01 by M V Berry, Marsha Berry, K E Mount +1 · 1,426 citations
Mathematics · Physics and Astronomy · #Action (physics) #Classical mechanics #Eigenfunction #Eigenvalues and eigenvectors #Limit (mathematics) #Mathematical analysis #Mathematical physics #Mathematics #Neutrino Physics Research #Path integral formulation #Physics #Quantum #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Quantum mechanics #Semiclassical physics #WKB approximation #Wave function

paper · pdf · doi:10.1088/0034-4885/35/1/306

published in Reports on Progress in Physics 35(1), 315-397 (IOP Publishing)

openalex publication_date 1972/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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. This review was completed in February 1972.

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