A class of exactly solved time-dependent quantum harmonic oscillators

Abstract

We consider a class of time-dependent harmonic oscillators, H(t)=p²/2mt^alpha + m omega ²t^bq²/2, whose mass and frequency vary as non-negative powers of time. Classically they describe damping oscillators slowly decaying as negative powers of time. Using the connection between classical and quantum harmonic oscillators we find analytically the Lewis-Riesenfeld invariants, obtain the exact quantum states, and compare these with the Caldirola-Kanai oscillator.