The extended Thomas-Fermi kinetic energy density functional with position-dependent effective mass in one dimension

The point canonical transformations map the Schrodinger equation with constant mass to a wave equation with a position-dependent effective mass. Using such a technique we derive, for a one-dimensional inhomogeneous system of noninteracting fermions with density p(x) and spatially dependent effective mass distribution m(x), the semiclassical kinetic energy density functional tau(p) in the so-called extended Thomas-Fermi model up to order h(2). For a given position-dependent mass, we compare numerically the total semiclassical kinetic energy with its exact quantum mechanical counterpart. The qualitative agreement is excellent.