The kinetic Hamiltonian with position-dependent mass

Abstract

In the present paper we examine in a systematic way the most relevant orderings of pure kinetic Hamiltonians for five different position-dependent mass (PDM) profiles: soliton-like, reciprocal quadratic and biquadratic, exponential and parabolic. As a result of the non-commutativity between momentum and position operators, a diversity of effective potentials is generated. We analyze the whole set and find unexpected coincidences as well as discrepancies among them. We obtain analytically the full-spectrum of energies and solutions in the twenty-five cases considered. It is shown how the simple ordinary constant-mass solutions are transformed into a variety of complex combinations of transcendental functions and arguments. We find that particles with a non-uniform mass density can present discrete energy spectra as well as continuous ones which can be bounded or not. These results are consistent with the fact that although the external potential is zero, PDM eigenfunctions are not actual free states but a sort of effective waves in a solid-state sample. This is precisely the origin of the position-dependent mass. In all the events we obtain exact complete spectral expressions. Our methodological procedure thus puts a wide diversity of Hamiltonian seeds on an equal footing in order to be compared. This allows choosing the better arrangement to model a specific solid or heterostructure once the spectrum of a given material is experimentally available. Finally, we perform a one-dimensional model calculation of a double heterostructure with a parabolic PDM particle in the interface region. Our study is also indicated for applications inside material structures with the addition of external potentials.