The Picard-Fuchs equation in classical and quantum physics: Application to higher-order WKB method

TL;DR

This paper demonstrates how the Picard-Fuchs equation can be used to efficiently compute higher-order WKB corrections in quantum mechanics, providing a new method for analyzing complex potentials.

Contribution

It introduces a novel application of the Picard-Fuchs equation to calculate higher-order WKB corrections for specific potentials, simplifying quantum correction computations.

Findings

Successfully calculated second-order WKB corrections for sextic and Lamé potentials.

Established a link between higher-order quantum corrections and derivatives of classical action.

Provided a self-contained guide for applying the Picard-Fuchs equation in quantum physics.

Abstract

The Picard-Fuchs equation is a powerful mathematical tool which has numerous applications in physics, for it allows to evaluate integrals without resorting to direct integration techniques. We use this equation to calculate both the classical action and the higher-order WKB corrections to it, for the sextic double-well potential and the Lam\'e potential. Our development rests on the fact that the Picard-Fuchs method links an integral to solutions of a differential equation with the energy as a parameter. Employing the same argument we show that each higher-order correction in the WKB series for the quantum action is a combination of the classical action and its derivatives. From this, we obtain a computationally simple method of calculating higher-order quantum-mechanical corrections to the classical action, and demonstrate this by calculating the second-order correction for the sextic and the Lam\'e potential. This paper also serves as a self-consistent guide to the use of the Picard-Fuchs equation.