On relativistic harmonic oscillator

Abstract

A relativistic quantum harmonic oscillator in 3+1 dimensions is derived from a quaternionic non-relativistic quantum harmonic oscillator. This quaternionic equation also yields the Klein-Gordon wave equation with a covariant (space-time dependent) mass. This mass is quantized and is given by $m_{\ast n}^2=m_{\omega }^2\left(n_r^2-1-\beta \left(n+1\right)\right),$ where $m_{\omega }=\frac{\hslash \omega }{c^2},$ $\beta =\frac{2m{c}^2}{\hslash \omega }$, $n$, is the oscillator index, and $n_r$ is the refractive index in which the oscillator travels. The harmonic oscillator in 3+1 dimensions is found to have a total energy of $E_{\ast n}=(n+1)\hslash \omega$, where $\omega$ is the oscillator frequency. A Lorentz invariant solution for the oscillator is also obtained. The time coordinate is found to contribute a term $-\frac{1}{2}\hslash \omega$ to the total energy. The squared interval of a massive oscillator (wave) depends on the medium in which it travels. Massless oscillators have null light cone. The interval of a quantum oscillator is found to be determined by the equation, $c^2{t}^2-r^2={\lambda }_c^2(1-n_r^2)$, where ${\lambda }_c$ is the Compton wavelength. The space-time inside a medium appears to be curved for a massive wave (field) propagating in it.

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References in corpus (1)

• On Interpretation of Special Relativity: a complement to Covariant Harmonic Oscillator Picture