The Quaternionic Quantum Mechanics

TL;DR

This paper develops a quaternionic formulation of quantum mechanics, deriving generalized wave equations and analyzing particle properties like spin, with results consistent with Einstein's energy-momentum relation.

Contribution

It introduces a quaternionic wavefunction framework that generalizes quantum wave equations and explores particle spin interactions within this new formalism.

Findings

Derived a quaternionic momentum eigenvalue equation.

Obtained generalized wave equations reducing to Klein-Gordon form.

Found complex angular frequency solutions consistent with energy-momentum relations.

Abstract

A quaternionic wavefunction consisting of real and scalar functions is found to satisfy the quaternionic momentum eigenvalue equation. Each of these components are found to satisfy a generalized wave equation of the form $\frac{1}{c^2}\frac{\partial^2\psi_0}{\partial t^2} - \nabla^2\psi_0+2(\frac{m_0}{\hbar})\frac{\partial\psi_0}{\partial t}+(\frac{m_0c}{\hbar})^2\psi_0=0$. This reduces to the massless Klein-Gordon equation, if we replace $\frac{\partial}{\partial t}\to\frac{\partial}{\partial t}+\frac{m_0c^2}{\hbar}$. For a plane wave solution the angular frequency is complex and is given by $\vec{\omega}_\pm=i\frac{m_0c^2}{\hbar}\pm c\vec{k} $, where $\vec{k}$ is the propagation constant vector. This equation is in agreement with the Einstein energy-momentum formula. The spin of the particle is obtained from the interaction of the particle with the photon field.