  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 hbar[(1+ beta p^2) delta_{ij} + beta' p_i
p_j]. These commutation relations are motivated by the fact 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.
