A new approach to solving the Schrödinger equation

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. The potential function defines the amplitude and the phase of any wavefunction which solves the one-dimensional Schrödinger equation. This new approach allows us to recover known solutions as well as to get new ones for both free and interacting particles with wavefunctions that have vanishing and non-vanishing Bohm potentials. For most of the potentials, no solutions to the Schrödinger 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 unexpected and surprising quantum results which seem to (but do not) violate the correspondence principle such as accelerated Airy wavefunctions which solve the free Schrödinger equation, are due to the presence of non-vanishing Bohm potentials. New examples of this kind are found and discussed. The relation of these results to some of the unusual solutions to other wave equations is briefly discussed.

Integrals whose saddle-point expansions terminate

“…Formulations in mathematical physics have occasionally been reverse-engineered to provide new solutions, or new insights into old ones. Examples are: determining potentials, or refractive-index profiles, that are reflectionless [1]; finding time-dependent Hamiltonians that do not induce transitions between eigenstates of a related system [2][3][4]; and getting new solutions of the Schrödinger equation by fixing the Madelung-Bohm potential [5].…”