Abstract
We introduce a class of potentials for which the time‐dependent Schrödinger equation with position‐dependent (effective) mass allows reduction to a stationary Schrödinger equation. This reduction is done by a particular point canonical transformation which preserves L2‐normalizability.
Highlights
Schrodinger equations with effective mass occur in the context of transport phenomena in crystals, where the electrons are not completely free, but interact with the potential of the lattice
The main problem of accessing time-dependent Schrodinger equations (TDSE) with effective mass is the lack of known solvable cases
In order to attack this problem for noneffective mass, it has been shown that for a certain class of potentials, the TDSE with constant mass can be mapped onto a stationary Schrodinger equation [14], such that each solvable stationary Schrodinger equation generates a solvable TDSE
Summary
Schrodinger equations with effective mass occur in the context of transport phenomena in crystals (e.g., semiconductors), where the electrons are not completely free, but interact with the potential of the lattice. Particular potentials with effective mass have been studied mainly by means of point canonical transformations [5–7] and Darboux transformations (resp., supersymmetric factorization) [8–10]. These methods have been elaborated for the fully timedependent case [11–13]. We identify a class of potentials for which the effective mass TDSE can be reduced to a stationary Schrodinger equation by means of a point canonical transformation. Each solvable stationary Schrodinger equation gives rise to a solvable effective mass TDSE This allows the straightforward generation of time-dependent potentials with effective masses and their corresponding solutions.
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