The quantum and Klein-Gordon oscillators in a non-commutative complex space and the thermodynamic functions

TL;DR

This paper investigates quantum and Klein-Gordon oscillators in non-commutative complex space, revealing their equations resemble those of particles with spin in magnetic fields and analyzing how non-commutativity influences thermodynamic properties.

Contribution

It introduces the study of oscillators in non-commutative complex space and derives their energy spectra and thermodynamic functions, highlighting the effects of non-commutativity.

Findings

Quantum oscillator equations resemble electron spin equations in magnetic fields.

Energy levels are exactly solvable in non-commutative space.

Non-commutativity influences energy at high temperatures.

Abstract

In this work we study the quantum and Klein-Gordon oscillators in non-commutative complex space. We show that the quantum oscillator in non-commutative complex space obeys an equation similar to the equation of motion of an electron with spin in a commutative space in an external uniform magnetic field. Therefore the wave function ${\psi}(z,\bar{z}) $ takes values in $C^{4}$, spin up, spin down, particle, antiparticle, a result which is obtained by the Dirac theory. We derive the thermodynamic functions associated to the partition function. We show that the non-commutativity affects energy at the high temperature limit. The Klein Gordon oscillator in non-commutative complex space also has a similar equation of motion to that a fermions with spin 1/2 in a commutative space in a constant magnetic field. The energy levels could be obtained by exact solution.