The Quaternionic Quantum Mechanics

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.

Note:

13 Latex pages, no figures

03.75.-b
03.65.Ge
03.65.Ca
03.65.Ta

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Mach's principle and a relativistic theory of gravitation

C. Brans
(
- Princeton U.
)
,
R.H. Dicke
(
- Princeton U.
)

- Phys.Rev. 124 (1961) 925-935,
•
DOI:
- 10.1103/PhysRev.124.925

- Phys.Rev.Lett. 93 046602