Simple variational approaches to quantum wells

TL;DR

This paper compares two simple variational methods for quantum wells, showing their effectiveness for different well depths and analyzing their series expansions and agreement with perturbation theory.

Contribution

It introduces and compares harmonic and exponential trial functions for variational calculations in quantum wells, highlighting their suitability for deep and shallow wells respectively.

Findings

Harmonic functions work well for deep wells.

Exponential functions are better for shallow wells.

Deep-well expansion matches perturbation theory results.

Abstract

We discuss two simple variational approaches to quantum wells. The trial harmonic functions analyzed in an earlier paper give reasonable results for all well depths and are particularly suitable for deep wells. On the other hand, the exponential functions proposed here are preferable for shallow wells. We compare the shallow-well expansions for both kind of functions and show that they do not exhibit the cubic term appearing in the exact series. It is also shown that the deep-well expansion for the harmonic functions agree with the first terms of perturbation theory.