Schr\"odinger equations with time-dependent P^2 and X^2 terms

Abstract

We present some general results for the time-dependent mass Hamiltonian problem with H=-{1/2}e^{-2\nu}\partial_{xx} +h^{(2)}(t)e^{2\nu}x^2. This Hamiltonian corresponds to a time-dependent mass (TM) Schr\"odinger equation with the restriction that there are only P^2 and X^2 terms. We give the specific transformations to a different quantum Schr\"odinger(TQ) equation and to a different time-dependent oscillator (TO) equation. For each Schr\"odinger system, we give the Lie algebra of space-time symmetries and (x,t) representations for number states, coherent states, and squeezed states. These general results include earlier work as special cases.