The Schrödinger system H=-{1/2}e^{Υ(t-t_o)}\partial_{xx} +\lfrac{1}{2}ω^2e^{-Υ(t-t_o)}x^2

Abstract

In this paper, we attack the specific time-dependent Hamiltonian problem H=-{1/2}e^{Υ(t-t_o)}\partial_{xx} +\lfrac{1}{2}ω^2e^{-Υ(t-t_o)}x^2. This corresponds to a time-dependent mass (TM) Schrödinger equation. We give the specific transformations to i) the more general quadratic (TQ) Schrödinger equation and to ii) a different time-dependent oscillator (TO) equation. For each Schrödinger system, we give the Lie algebra of space-time symmetries, the number states, the coherent states, the squeezed-states and the time-dependent <x>, <p>, (Δx)^2, (Δp)^2, and uncertainty product.