Exact general solution of the Schrödinger equation by modified Cole-Hopf transformation

Alan Clark Longfellow

Published Dec 9, 2024• 0 citations

Abstract

The Schrödinger equation is a linear partial differential equation that describes the temporal evolution of the state of a quantum mechanical system. One of the pillars of quantum mechanics, the Schrödinger equation has been used to describe a wide variety of phenomena such as wave-particle duality, atomic orbitals, quantum oscillations, and even some fluid mechanics problems. Despite its established importance in the development of quantum mechanics, a solution of the Schrödinger equation for an arbitrary distribution of quantum mechanical potential or for an arbitrary number of space dimensions remains unknown. This work seeks to rectify this theoretical roadblock by providing an analytical solution through a recasting of the well-known Cole-Hopf transformation which is used to linearize a broad class of PDEs of which the Schrödinger equation becomes a member after an earlier change of variables. These two successive transformations render the general wave function proportional to the solution of a forced heat equation, allowing an explicit analytical solution applicable to any system in any number of space dimensions. Two practical applications to deriving the explicit wave functions of multielectronic atoms and to the forced Burgers equation of fluid mechanics are discussed, enabling greater mathematical insight into quantum entanglement and nonlinear fluid systems.

DOI: 10.31219/osf.io/4gvzt

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วธการทางคณตศาสตรสำหรบสมการชเรอดงเงอร และวธประมาณคาแบบดบเบลยเคบ

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