Equispaced-level Hamiltonians with the variable effective mass following the potential

A coordinate-transform-based procedure is presented for generating Hamiltonians with the effective-mass variation proportional to the potential variation, and having equispaced bound states. A situation like that occurs in graded semiconductor quantum wells based on ternary alloys, e.g., ${\mathrm{Al}}_{\mathit{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As, and the procedure aims at designing the structure to have equispaced states. Two types of Hamiltonians are derived, one with the confining potential that may be realized by appropriate grading of the alloy, and another with a nonconfining potential but with the effective mass tending to zero at infinity. This latter type of Hamiltonian is not realizable in semiconductor structures, but gives insight into the nature of wave-function confinement, which makes it interesting in its own right. © 1996 The American Physical Society.