Quantum harmonic oscillator for dipoles

TL;DR

This paper explores the conditions under which bound states occur for neutral particles with dipole moments in radial fields, revealing that specific field configurations and non-commutative geometry influence the energy spectrum.

Contribution

It demonstrates that Landau analog levels require radial fields proportional to the cube of distance and shows non-commutative phase space can produce harmonic spectra without external fields.

Findings

Bound states exist only with cubic radial field dependence.

Non-commutative geometry induces harmonic spectra without external fields.

Wave functions and energy levels are derived for both commutative and non-commutative cases.

Abstract

In this work we investigate which radial field configuration yields bound states for neutral particles showing non-zero magnetic and electric dipole moments. The main result is that, in contrast with previous works, the Landau analog levels only exist if these radial magnetic and electric external fields are proportional to the third power of distance, not proportional to the distance. We derive the wave functions and the energy levels in the context of commutative and non-commutative quantum mechanics. We also show that, in the case of non-commutative phase space, these harmonic oscillator like spectrum do exist even if there is no external radial magnetic and electric fields. They are only consequence of the non-commutativity in the momenta.