Operator ordering in effective-mass theory for heterostructures. I. Comparison with exact results for superlattices, quantum wells, and localized potentials

We study, for heterostructures with abrupt interfaces, the problem of operator ordering in the effective-mass Hamiltonian with kinetic-energy operator 1/2${\mathit{m}}^{\mathrm{\alpha }}$p${\mathit{m}}^{\mathrm{\beta }}$p${\mathit{m}}^{\mathrm{\alpha }}$, for a position-dependent effective mass. Here, 2α+β=-1. Through exact model calculations on superlattices, quantum wells, and localized potentials we show that when effective-mass theory is applicable, α=0 and β=-1. In all cases the effective-mass theory has the status of an asymptotically exact treatment, except for strained lattices.