The Modified Airy Function Approximation Applied to the Double-Well Potential

TL;DR

This paper compares the WKB and Modified Airy Function (MAF) approximations for quantum potentials, demonstrating MAF's superior accuracy in eigenvalues and wave functions, and advocates for its inclusion in undergraduate teaching.

Contribution

It introduces the MAF approximation as a better alternative to WKB for double-well potentials, providing accurate eigenvalues and wave functions.

Findings

MAF yields more accurate wave functions than WKB.

Eigenvalues obtained via MAF closely match exact solutions.

MAF improves understanding of quantum tunnelling and superposition.

Abstract

The single harmonic oscillator and double-well potentials are important systems in quantum mechanics. The single harmonic oscillator is {\it the} paradigm in physics, and is taught in nearly all beginner undergraduate classes, while the double-well potential illustrates the two important principles of quantum tunnelling and linear superposition. While exact analytical solutions of the Schrodinger equation exist for both of these potentials, they are also employed to benchmark the use of approximate techniques which may be the only recourse for more complicated potentials. In this paper, we review the Wentzel-Kramers-Brillouin (WKB) approximation for both these potentials. While this approximation is known for its accurate energies, we will instead emphasize how poor the WKB wave functions are. The inaccuracy of the WKB wave functions will then motivate us to adopt the lesser-known Modified Airy Function (MAF) approximation, which alleviates the deficiencies of the WKB wave functions. We will review the MAF solution to the simple harmonic oscillator potential, and then apply the MAF to the double-well potential. We find accurate eigenvalues and, more importantly, very accurate wave functions. We conclude with the suggestion that an introduction to the MAF should be included in undergraduate courses to complement the WKB.