We have derived an analytical solution of the 3D Schrequation where the mass varies with position. Our solution is a general solution for any massive particle with a position dependent mass m(r) scattered by an arbitrary potential V(r)
Using a recently developed technique to solve Schrödinger equation for constant mass, we studied the regime in which mass varies with position i.e position dependent mass Schrödinger equation(PDMSE). We obtained an analytical solution for the PDMSE and applied our approach to study a position dependent mass m(x) particle scattered by a potential V(x). We also studied the structural analogy between PDMSE and two-level atomic system interacting with a classical field.
An algebraic method of constructing potentials for which the Schrödinger equation with position dependent mass can be solved exactly is presented. A general form of the generators of su (1,1) algebra has been employed with a unified approach to the problem. Our systematic approach reproduces a number of earlier results and also leads to some novelties. We show that the solutions of the Schrödinger equation with position dependent mass are free from the choice of parameters for position dependent mass. Two classes of potentials are constructed that include almost all exactly solvable potentials.
In the framework of the position-dependent mass quantum mechanics, the three dimensional Schrödinger equation is studied by applying the Laplace transforms combining with the point canonical transforms. For the potential analogues to Morse potential and via the Pekeris approximation, we introduce the general solutions appropriate for any kind of position dependent mass profile which obeys a key condition. For a specific position-dependent mass profile, the bound state solutions are obtained through an analytical form. The constant mass solutions are also relived.