A new approach to solving the Schr\"odinger equation

TL;DR

This paper introduces a novel method for solving the one-dimensional Schrödinger equation by using a potential function for the wavefunction, enabling the discovery of both known and new solutions, especially highlighting the role of Bohm potentials.

Contribution

The paper proposes a new approach based on a potential function for the wavefunction, identifying conditions for solutions with vanishing Bohm potential and providing new examples of quantum states.

Findings

Identified a family of potentials with solutions having vanishing Bohm potential

Reproduced known solutions and found new solutions for free and interacting particles

Demonstrated the significance of non-vanishing Bohm potentials in quantum phenomena

Abstract

A new approach to find exact solutions to one--dimensional quantum mechanical systems is devised. The scheme is based on the introduction of a potential function for the wavefunction, and the equation it satisfies. We recover known solutions as well as to get new ones for both free and interacting particles with wavefunctions having vanishing and non--vanishing Bohm potentials. For most of the potentials, no solutions to the Schr\"odinger equation produce a vanishing Bohm potential. A (large but) restricted family of potentials allows the existence of particular solutions for which the Bohm potential vanishes. This family of potentials is determined, and several examples are presented. It is shown that some quantum, such as accelerated Airy wavefunctions, are due to the presence of non--vanishing Bohm potentials. New examples of this kind are found and discussed.