How to derive the Schrodinger equation for a system with position dependent effective mass? For example, I encountered this equation when I first studied semiconductor hetero-structures. All the books that I referred take the equation as the starting point without at least giving an insight into the form of the Schrodinger equation which reads as
$$\big[-\frac{\hbar^2}{2}\nabla \frac{1}{m^*}\nabla + U \big]\Psi ~=~ E \Psi. $$
I feel that it has something to do with conserving the probability current and should use the continuity equation, but I am not sure.
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