# On relativistic harmonic oscillator. (arXiv:1709.06865v1 [physics.gen-ph])

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

$m_{*n}^2=m_\omega^2\left(n_r^2-1-\beta\,\left(n+1\right)\right)\,,$ where

$m_\omega=\frac{\hbar\omega}{c^2}\,,$ $\beta=\frac{2mc^2}{\hbar\,\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 $E_{*n}=(n+1)\,\hbar\,\omega$, 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

$-\frac{1}{2}\,\hbar\,\omega$ 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, $c^2t^2-r^2=\lambda^2_c(1-n_r^2)$, 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.