Wave function for smooth potential and mass step

The one-dimensional Schr\"odinger equation, derived from the general form of the effective-mass Hamiltonian ${(m}^{\ensuremath{\eta}}\mathbf{p}{m}^{\ensuremath{\epsilon}}\mathbf{p}{m}^{\ensuremath{\rho}}{+m}^{\ensuremath{\rho}}\mathbf{p}{m}^{\ensuremath{\epsilon}}\mathbf{p}{m}^{\ensuremath{\eta}})/4+V$ with $\ensuremath{\eta}+\ensuremath{\epsilon}+\ensuremath{\rho}=\ensuremath{-}1,$ is solved exactly for a system with smooth potential and mass step. The wave function depends on the Heun function, which is a generalization of the hypergeometric function. The effective-mass Hamiltonian and the connection rules for a system with abrupt heterojunction are deduced from the study of the limiting case when the smooth step potential and mass tend to an abrupt potential and mass step.