The relativistic two-dimensional harmonic oscillator

TL;DR

This paper introduces an exactly solvable relativistic 2D harmonic oscillator model in configurational space, deriving wave functions, energy spectrum, and momentum operator, with results consistent with non-relativistic limits.

Contribution

It presents the first exactly solvable finite-difference model of a relativistic 2D harmonic oscillator in configurational space, including explicit wave functions and operators.

Findings

Wave functions and energy spectrum are explicitly derived.

Model reduces correctly to non-relativistic harmonic oscillator.

Explicit form of the plane momentum operator is obtained.

Abstract

The two-dimensional relativistic configurational $\vec r$-space is proposed and the exactly solvable finite-difference model of the harmonic oscillator in this space is constructed. The wave functions of the stationary states and the corresponding energy spectrum are found for the model under consideration. It is shown that, they have correct non-relativistic limit. Explicit form of the plane momentum operator is determined.