The Step-Harmonic Potential

TL;DR

This paper investigates a quantum system with a step plus harmonic barrier potential, solving the eigenvalue problem and analyzing wave packet reflection to connect quantum interaction times with classical limits.

Contribution

It introduces a novel integral representation method to classify solutions and analyzes wave packet dynamics in a combined step-harmonic potential.

Findings

Eigenvalue solutions classified via complex plane paths

Interaction time matches classical half-period at high energies

Method provides insights into quantum-classical correspondence

Abstract

We analyze the behavior of a quantum system described by a one-dimensional asymmetric potential consisting of a step plus a harmonic barrier. We solve the eigenvalue equation by the integral representation method, which allows us to classify the independent solutions as equivalence classes of homotopic paths in the complex plane. We then consider the propagation of a wave packet reflected by the harmonic barrier and obtain an expression for the interaction time as a function of the peak energy. For high energies we recover the classical half-period limit.