WKB Approximation to the Power Wall

TL;DR

This paper develops a semiclassical approach to analyze quantum particles near steep potential walls, comparing exact solutions with approximations, classifying classical paths, and testing convergence hypotheses in the presence of caustics.

Contribution

It introduces a semiclassical propagator for steep potentials, classifies classical paths, and tests convergence hypotheses, extending understanding of quantum boundary problems.

Findings

Semiclassical propagator matches exact solutions for specific potentials.

Classifies classical paths and calculates their actions and amplitudes.

Identifies limitations of hypotheses when caustics form.

Abstract

We present a semiclassical analysis of the quantum propagator of a particle confined on one side by a steeply, monotonically rising potential. The models studied in detail have potentials proportional to $x^{\alpha}$ for $x>0$; the limit $\alpha\to\infty$ would reproduce a perfectly reflecting boundary, but at present we concentrate on the cases $\alpha =1$ and 2, for which exact solutions in terms of well known functions are available for comparison. We classify the classical paths in this system by their qualitative nature and calculate the contributions of the various classes to the leading-order semiclassical approximation: For each classical path we find the action $S$, the amplitude function $A$ and the Laplacian of $A$. (The Laplacian is of interest because it gives an estimate of the error in the approximation and is needed for computing higher-order approximations.) The resulting semiclassical propagator can be used to rewrite the exact problem as a Volterra integral equation, whose formal solution by iteration (Neumann series) is a semiclassical, not perturbative, expansion. We thereby test, in the context of a concrete problem, the validity of the two technical hypotheses in a previous proof of the convergence of such a Neumann series in the more abstract setting of an arbitrary smooth potential. Not surprisingly, we find that the hypotheses are violated when caustics develop in the classical dynamics; this opens up the interesting future project of extending the methods to momentum space.