Formulation of the Relativistic Harmonic Oscillator in terms of Quaternions

Abstract

A relativistic harmonic oscillator is formulated in terms of quaternions derived from the non-relativistic quantum harmonic oscillator. The total energy is found to be E = (n + 1) ħω, the three spacial degrees of freedom contributed an amount of energy equals to (n+ 3/"2" ) ħω, whereas the time component contributed an amount of (-"ħ" ω)/"2" . The quaternionic wavefunction represents a state of a particle with orbital angular momentum, L"0" = 3/2 "ħ" , resulting from the internal structure of the particle. The commutation relations between the quaternionic position and momentum operators with the quaternionic Hamiltonian are reducing to that of ordinary operators. In presence of an electromagnetic field, the vector wavefunction is found to have a physical meaning related to the particle spin. The quaternionic continuity equation yield two equations provide that the probability density ρ and the current density J ⃗ are due to composed particles of mass M equals twice the mass of the particle, which can be useful in describing the superconductivity.