Yet another position-dependent mass quantum model

Abstract

The quantum dynamics of particles with mass dependent on the position is a problem of interest since the effective-mass approach to charge carriers in conductors and semiconductors began to be used. These problems have been solved using the Hamiltonian H = 1 2mα(x)pmβ (x)pmα(x), where α and β are real parameters which satisfy the condition 2α + β = − 1. It has been verified that the choice α = 0, β = − 1 is compatible with Galilean invariance. In this work we propose a new Hamiltonian, Hˆ = 1 6

mˆ (xˆ) −1 pˆ2 + pˆmˆ (xˆ) −1 pˆ + p2mˆ (xˆ) −1

, to describe variable mass systems. We considered every permutation among the operators, taking into account that the mass is now an operator. We verified that this Hamiltonian is Hermitian and is compatible with Galilean invariance. For comparison, we used both Hamiltonians to calculate the band structure for a quantum particle with mass varying periodically. Although qualitatively equivalent, the results turn out to produce different numerical values.