Energy Evolution in Time-Dependent Harmonic Oscillator

Abstract

The theory of adiabatic invariants has a long history, and very important implications and applications in many different branches of physics, classically and quantally, but is rarely founded on rigorous results. Here we treat the general time-dependent one-dimensional harmonic oscillator, whose Newton equation [Formula: see text] cannot be solved in general. We follow the time-evolution of an initial ensemble of phase points with sharply defined energy E0at time t = 0 and calculate rigorously the distribution of energy E1after time t = T, which is fully (all moments, including the variance μ2) determined by the first moment [Formula: see text]. For example, [Formula: see text], and all higher even moments are powers of μ2, whilst the odd ones vanish identically. This distribution function does not depend on any further details of the function ω(t) and is in this sense universal. In ideal adiabaticity [Formula: see text], and the variance μ2is zero, whilst for finite T we calculate [Formula: see text], and μ2for the general case using exact WKB-theory to all orders. We prove that if ω(t) is of class [Formula: see text] (all derivatives up to and including the order m are continuous) μ ∝ T-(m+1), whilst for the class [Formula: see text] it is known to be exponential μ ∝ exp (-αT).