The double well potential in quantum mechanics: a simple, numerically exact formulation

TL;DR

This paper presents a simple, numerically exact method for solving the double well potential in quantum mechanics, demonstrating high accuracy of the WKB approximation across multiple states, making it accessible for undergraduates.

Contribution

It introduces an accessible numerical approach for exact eigenenergies and eigenstates of the double well potential, comparing results with the WKB approximation.

Findings

Numerically exact solutions are achievable with elementary mathematics.

WKB approximation is highly accurate for multiple energy states.

Method makes double well potential solutions accessible to undergraduates.

Abstract

The double well potential is arguably one of the most important potentials in quantum mechanics, because the solution contains the notion of a state as a linear superposition of `classical' states, a concept which has become very important in quantum information theory. It is therefore desirable to have solutions to simple double well potentials that are accessible to the undergraduate student. We describe a method for obtaining the numerically exact eigenenergies and eigenstates for such a model, along with the energies obtained through the Wentzel-Kramers-Brillouin (WKB) approximation. The exact solution is accessible with elementary mathematics, though numerical solutions are required. We also find that the WKB approximation is remarkably accurate, not just for the ground state, but for the excited states as well.