Quaternionic quantum harmonic oscillator

TL;DR

This paper derives solutions for the quantum harmonic oscillator within quaternionic quantum mechanics, revealing additional possibilities over traditional complex quantum mechanics and opening avenues for future research.

Contribution

It provides the first harmonic oscillator solutions in quaternionic quantum mechanics using both analytic and algebraic methods.

Findings

Quaternionic solutions extend beyond complex quantum mechanics.

Multiple solution methods demonstrated.

Potential applications in future quantum research.

Abstract

In this article we obtained the harmonic oscillator solution for quaternionic quantum mechanics ($\mathbbm{H}$QM) in the real Hilbert space, both in the analytic method and in the algebraic method. The quaternionic solutions have many additional possibilities if compared to complex quantum mechanics ($\mathbbm{C}$QM), and thus there are many possible applications to these results in future research.