Exactly solvable time-dependent problems: Potentials of monotonously decreasing function of time

Abstract

We solve the Schrödinger equation analytically for systems whose potentials have a certain time-dependence (which is monotonously decreasing) and general coordinate-dependences. Only a few time-dependent systems have been reported to be analytically solved whose potentials are constant, linear, and quadratic functions of coordinate with arbitrary time-dependences. From a different perspective, we focus on the time-dependent systems whose potentials are monotonously decreasing functions of time with arbitrary coordinate-dependences. Time-dependent potential of any coordinate-dependence can be handled analytically by transforming it to a time-independent potential of known solutions if its time-dependence is monotonously decreasing. We do this by a unitary transformation of the wavefunction and variable transformations to change the Schrödinger equation to be time-independent in new variables. These variables are then determined by solving a set of simple differential equations. This way we are able to find and to obtain analytical solutions for time-dependent potentials which we mention above.