Isomorphism of Analytical Spectrum between Noncommutative Harmonic Oscillator and Landau Problem

TL;DR

This paper demonstrates the conditions under which the noncommutative harmonic oscillator and Landau problem are spectrally and structurally equivalent, revealing their isomorphism and the effects of noncommutativity and magnetic fields.

Contribution

It analytically compares the Hamiltonians and eigenstates of both systems, establishing isomorphism conditions and exploring the impact of noncommutativity and gauge choices.

Findings

Systems are isomorphic under specific quantum number and magnetic field conditions.

Landau gauge requires loss of one spatial degree of freedom for isomorphism.

Noncommutativity influences eigenstates and probability distributions.

Abstract

The comparison of the Hamiltonians of the noncommutative isotropic harmonic oscillator and Landau problem are analysed to study the specific conditions under which these two models are indistinguishable. The energy eigenvalues and eigenstates of Landau problem in symmetric and two Landau gauges are evaluated analytically. The Hamiltonian of a noncommutative isotropic harmonic oscillator is found by using Bopp's shift in commutative coordinate space. The result shows that the two systems are isomorphic up to the similar values of $n_{r}$ and $m_{l}$ and $qB = eB > 0$ for both gauge choices. However, there is an additional requirement for Landau gauge where the noncommutative oscillator has to lose one spatial degree of freedom. It also needs to be parametrized by a factor $\zeta$ for their Hamiltonians to be consistent with each other. The wavefunctions and probability density functions are then plotted and the behaviour that emerges is explained. Finally, the effects of noncommutativity or magnetic field on the eigenstates and their probability distribution of the isomorphic system are shown.