Reduction of Time-Dependent Schrödinger Equations with Effective Mass to Stationary Schrödinger Equations

Abstract

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. The quantum dynamics of such electrons can be modeled by an effective mass, the behavior of which is determined by the band curvature [1–3]. Since the effective mass Schrodinger equation takes a more complicated form than the conventional Schrodinger equation [4], the identification of solvable cases is more difficult. In the stationary case, 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]. Recently, these methods have also been elaborated for the fully timedependent case [11–13]. However, 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 (constant) 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. Whereas in [14] the most general class of reducible potentials is derived, particular cases have been obtained earlier, for example, for time-dependent harmonic oscillator potentials with travelling-wave terms [15]. The method of mapping time-dependent onto stationary problems has also been used for the calculation of Green’s functions for timedependent Coulomb and other potentials [16]. The purpose of the present note is to generalize the results in [14] to the effective mass case. 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. Thus, 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. Furthermore, our transformation preserves L2-normalizability, such that physical solutions are taken into physical solutions. In the remainder of this note, we first give the point canonical transformation that relates effective mass TDSEs and stationary Schrodinger equations to each other. We prove its correct simplification for constant mass and show that L2-normalizability of the solutions is preserved. In the final paragraph we present an application.