Form-preserving Transformations for the Time-dependent Schrodinger Equation in (' + 1) Dimensions

摘要

We define a form-preserving transformation (also called point canonical transformation) for the time-dependent SchrSdinger equation (TDSE) in (n + 1) dimensions. The form-preserving transformation is shown to be invertible and to preserve L2-normalizability. We give a class of time-dependent TDSEs that can be mapped onto stationary SchrSdinger equations by our form-preserving transformation. As an example, we generate a solvable, time-dependent potential of Coulombic ring-shaped type together with the corresponding exact solution of the TDSE in (3+1) dimensions. We further consider TDSEs with position-dependent (effective) masses and show that there is no form-preserving transformation between them and the conventional TDSEs, if the spatial dimension of the system is higher than one.