Tunneling in cosine potential with periodic boundary conditions

TL;DR

This paper compares three methods—Bloch waves, semi-classical approximation, and numerical basis approach—for calculating energy splitting in a cosine potential with periodic boundary conditions, highlighting their accuracy and limitations.

Contribution

It introduces and evaluates three distinct methods for computing energy splitting in cosine potentials, emphasizing the improved accuracy of the numerical approach.

Findings

Numerical approach provides the most precise results.

WKB method helps determine unknown coefficients in semi-classical approximation.

Bloch wave method offers only qualitative insights.

Abstract

In this paper we discuss three methods to calculate energy splitting in cosine potential on a circle, Bloch waves, semi--classical approximation and restricted basis approach. While the Bloch wave method gives only a qualitative result, with the WKB method we are able to determine its unknown coefficients. The numerical approach is most exact and enables us to extract further corrections to previous results.