The Quaternionic Quantum Mechanics

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 $\frac{1}{c^2}\frac{{\partial }^2{\psi }_0}{\partial t^2}-{\nabla }^2{\psi }_0+2(\frac{m_0}{\hslash })\frac{\partial {\psi }_0}{\partial t}+(\frac{m_{0}c}{\hslash }{)}^2{\psi }_0=0$. This reduces to the massless Klein-Gordon equation, if we replace $\frac{\partial }{\partial t}\to \frac{\partial }{\partial t}+\frac{m_0{c}^2}{\hslash }$. For a plane wave solution the angular frequency is complex and is given by ${\vec{\omega }}_{\pm }=i\frac{m_0{c}^2}{\hslash }\pm c\vec{k}$, 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.

13 Latex pages, no figures

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