A Solution Technique for Quantum Mechanical Differential Equations Using Multiple Complex Planes

TL;DR

This paper introduces a novel method for solving quantum mechanical differential equations by decomposing fields into multiple complex planes, demonstrated through the Schrödinger equation for hydrogen in a multi-plane complex space.

Contribution

It presents a new approach to represent quantum fields over multiple complex planes, enabling solutions to be holomorphic in these planes, expanding the mathematical tools for quantum mechanics.

Findings

Fields not representable over a single complex plane can be decomposed into multiple planes.

Eigensolutions of the Schrödinger equation are holomorphic in the complex planes.

The method successfully solves the hydrogen atom Schrödinger equation in a multi-plane complex space.

Abstract

It is shown fields that cannot be represented over one complex plane can be further decomposed for representation over multiple complex planes. This finding is demonstrated here by solving of the Schr\"{o}dinger equation for the hydrogen atom in a complex space containing three complex planes. The complex coordinate system is generated from real coordinates using an isometric transformation. One plane is applied to mix energy and time; the other two planes are used to represent the z-component of angular momentum of the electron. The eigensolutions of the Schr\"{o}dinger equation are shown to be holomorphic in the complex planes.