Analytical spectrum isomorphism of noncommutative harmonic oscillator and charged particle in magnetic field

The comparison of the Hamiltonians of a noncommutative isotropic harmonic oscillator and a charged particle in a uniform magnetic field in symmetric and Landau gauges are analysed to study the specific conditions under which these two models are indistinguishable. The Hamiltonian of the noncommutative isotropic harmonic oscillator is stated by applying Bopp shift transformation. The result shows that the two systems are isomorphic up to the similar values of quantum numbers nr and ml with the product of electric charge and magnetic field to be qb = eb > 0 for both gauge choices respectively. In addition, there is an extra requirement for Landau gauge where the noncommutative oscillator has to lose a single spatial degree of freedom. It also needs to be parametrized by a factor ζ to be determined for their Hamiltonians to be consistent with each other.

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