Form of the quantum kinetic-energy operator with spatially varying effective mass

Through a nonrelativistic limit of the Dirac Hamiltonian and considering the spatial dependence of the particle mass in the Foldy-Wouthuysen transformation, it is shown that the quantum kinetic-energy operator with spatially varying effective mass has the form |P[p-hat[1/√m(r)]p-hat[1/√m(r)]+[1/√m(r)]p-hat[1/√m(r)]p-hat|P]. However, the transmission properties of an electron through a GaAs/Alₓ Ga_1-x As heterojunction calculated both with this form and the BenDaniel and Duke operator p-hat[m(r)]^-1p-hat/2, the most used kinetic-energy operator, differ at most by 1% if the existence of interface regions as thin as two GaAs lattice units is taken into account.