The quantum harmonic oscillator on a lattice

The authors find the eigenvalue spectrum of a particle constrained to 'hop' between the sites of a simple cubic lattice in the presence of a spherically asymmetric 'harmonic oscillator' potential. The eigenfunctions are found to be given in terms of the periodic Mathieu functions of period pi . At low kinetic energies of hopping the solutions differ considerably from those of the continuum theory: the particle localises itself in a given shell around the origin and the virial theorem is thereby violated. At high kinetic energies they obtain the usual equidistant levels of the harmonic oscillator on the continuous manifold. They discuss in some detail the spectrum of the three-dimensional case and the associated degeneracies. It is seen that the well-known accidental degeneracy of the harmonic oscillator wavefunctions is only partially lifted by the discretisation of space. Finally, mention is also made of the relevance of this solution to the physical problem of the inversion layer at the surface of a semiconductor when it is doped with a spatially constant electric charge density.