A large class of bound-state solutions of the Schr\"odinger equation via Laplace transform of the confluent hypergeometric equation

TL;DR

This paper demonstrates a method using Laplace transforms of the confluent hypergeometric equation to analytically solve for bound states in a broad class of quantum systems, expressing solutions with Laguerre polynomials.

Contribution

It introduces a novel analytical approach to solve the Schrödinger equation for various systems, expanding the set of solvable potentials in quantum mechanics.

Findings

Bound states can be obtained analytically using Laplace transforms.

Eigenfunctions are expressed in terms of generalized Laguerre polynomials.

The method is illustrated with the generalized Morse potential.

Abstract

It is shown that analytically soluble bound states of the Schr\"odinger equation for a large class of systems relevant to atomic and molecular physics can be obtained by means of the Laplace transform of the confluent hypergeometric equation. It is also shown that all closed-form eigenfunctions are expressed in terms of generalized Laguerre polynomials. The generalized Morse potential is used as an illustration.