The Schr\"odinger system H=-{1/2} (t_o/t)^a \partial_{xx} + (1/2) \omega^2 (t/t_o)^b x^2

Abstract

We attack the specific time-dependent Hamiltonian problem H=-{1/2} (t_o/t)^a \partial_{xx} + (1/2) \omega^2 (t/t_o)^b x^2. This corresponds to a time-dependent mass (TM) Schr\"odinger equation. We give the specific transformations to a different time-dependent quadratic Schr\"odinger equations (TQ) and to a different time-dependent oscillator (TO) equation. For each Schr\"odinger system, we give the Lie algebra of space-time symmetries, the number states, the squeezed-state <x> and <p> (with their classical motion), (\Delta x)^2, (\Delta p)^2, and the uncertainty product.