The Laplace method for energy eigenvalue problems in quantum mechanics

TL;DR

This paper introduces a Laplace method-based approach for solving quantum energy eigenvalue problems, offering an alternative to traditional series solutions for both bound and continuum states, applicable to potentials with hypergeometric function solutions.

Contribution

It presents a novel application of the Laplace method to quantum eigenvalue problems, expanding the toolkit beyond Frobenius series solutions for exactly solvable potentials.

Findings

The Laplace method can solve Schrödinger equations with hypergeometric solutions.

It provides a unified approach for bound and continuum states.

The method is historically rooted and pedagogically valuable.

Abstract

Quantum mechanics has about a dozen exactly solvable potentials. Normally, the time-independent Schroedinger equation for them is solved by using a generalized series solution for the bound states (using the Froebenius method) and then an analytic continuation for the continuum states (if present). In this work, we present an alternative way to solve these problems, based on the Laplace method. This technique uses a similar procedure for the bound states and for the continuum states. It was originally used by Schroedinger when he solved for the wavefunctions of hydrogen. Dirac advocated using this method too. We discuss why it is a powerful approach for graduate students to learn and describe how it can be employed to solve all problems whose wavefunctions are represented in terms of confluent hypergeometric functions.