Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
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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 q + 驴2(t)q = 0 cannot be solved in general. We follow the time-evolution of an initial ensemble of phase points with sharply defined energy Eo at time t = 0 and calculate rigorously the distribution of energy E1 after time t = T, which is fully (all moments, including the variance 驴2) determined by the first moment 驴1. For example, 驴2 = E2o[(驴1/Eo)2 -- (驴(T)/驴(0))2]/2, and all higher even moments are powers of 驴2, whilst the odd ones vanish identically. This distribution function does not depend on any further details of the function 驴(t) and is in this sense universal. In ideal adiabaticity 驴1 = 驴(T)Eo/驴(0), and the variance 驴,2 is zero, whilst for finite T we calculate 驴1, and 驴2 for the general case using exact WKB-theory to all orders. We prove that if 驴(t)is of class $${\cal C}$$ m (all derivatives up to and including the order m are continuous) 驴, 驴 T驴(m + 1)) whilst for the class $${\cal C}$$ °° it is known to be exponential 驴 驴 exp(--aT).