Exact solution of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relations

Abstract

We determine the energy eigenvalues and eigenfunctions of the harmonic oscillator where the coordinates and momenta are assumed to obey the modified commutation relations $[{x}_{i},{p}_{j}]=i\ensuremath{\Elzxh}[(1+\ensuremath{\beta}{p}^{2}){\ensuremath{\delta}}_{\mathrm{ij}}+{\ensuremath{\beta}}^{\ensuremath{'}}{p}_{i}{p}_{j}].$ These commutation relations are motivated by the fact that they lead to the minimal length uncertainty relations which appear in perturbative string theory. Our solutions illustrate how certain features of string theory may manifest themselves in simple quantum mechanical systems through the modification of the canonical commutation relations. We discuss whether such effects are observable in precision measurements on electrons trapped in strong magnetic fields.