Separation of Variables and Exactly Soluble Time-Dependent Potentials in Quantum Mechanics

TL;DR

This paper investigates the conditions under which separation of variables can be used to find exactly solvable time-dependent potentials in quantum mechanics, revealing that such models are highly restricted and applying the findings to propagator calculations.

Contribution

It provides a formal analysis of the limitations of separation of variables for time-dependent quantum potentials and links these results to existing research and practical propagator computations.

Findings

Separation of variables applies only to a restricted class of time-dependent models.

Exact solubility requires specific transformations of coordinates and wavefunctions.

The methods are applied to compute quantum propagators.

Abstract

We use separation of variables as a tool to identify and to analyze exactly soluble time-dependent quantum mechanical potentials. By considering the most general possible time-dependent re-definition of the spatial coordinate, as well as general transformations on the wavefunctions, we show that separation of variables applies and exact solubility occurs only in a very restricted class of time-dependent models. We consider the formal structure underlying our findings, and the relationship between our results and other work on time-dependent potentials. As an application of our methods, we apply our results to the calculations of propagators.