The Stark effect in linear potentials

TL;DR

This paper derives exact closed-form expressions for the Stark effect in two one-dimensional linear potential quantum systems, comparing these results with approximation methods to enhance understanding of the Stark shift phenomena.

Contribution

It provides the first closed-form solutions for the Stark effect in linear potential models and compares them with perturbation and WKB approximations.

Findings

Exact closed-form Stark shifts derived for quantum bouncer and symmetric linear potential.

Comparison shows the accuracy of approximation methods against exact solutions.

Enhances analytical understanding of Stark effect in simple quantum systems.

Abstract

We examine the Stark effect (the second-order shift in the energy spectrum due to an external constant force) for two 1-dimensional model quantum mechanical systems described by linear potentials, the so-called quantum bouncer (defined by V(z) = Fz for z>0 and V(z) infinite for z<0) and the symmetric linear potential (given by V(z) = F|z|). We show how straightforward use of the most obvious properties of the Airy function solutions and simple Taylor expansions give closed form results for the Stark shifts in both systems. These exact results are then compared to other approximation techniques, such as perturbation theory and WKB methods. These expressions add to the small number of closed-form descriptions available for the Stark effect in model quantum mechanical systems.