  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 $\ddot{q} + \omega^2(t) q=0$ cannot be solved in general. We
follow the time-evolution of an initial ensemble of phase points with sharply
defined energy $E_0$ at time $t=0$ and calculate rigorously the distribution of
energy $E_1$ after time $t=T$, which is fully (all moments, including the
variance $\mu^2$) determined by the first moment $\bar{E_1}$. For example,
$\mu^2 = E_0^2 [(\bar{E_1}/E_0)^2 - (\omega (T)/\omega (0))^2]/2$, and all
higher even moments are powers of $\mu^2$, whilst the odd ones vanish
identically. This distribution function does not depend on any further details
of the function $\omega (t)$ and is in this sense universal. In ideal
adiabaticity $\bar{E_1} = \omega(T) E_0/\omega(0)$, and the variance $\mu^2$ is
zero, whilst for finite $T$ we calculate $\bar{E_1}$, and $\mu^2$ for the
general case using exact WKB-theory to all orders. We prove that if $\omega
(t)$ is of class ${\cal C}^{m}$ (all derivatives up to and including the order
$m$ are continuous) $\mu \propto T^{-(m+1)}$, whilst for class ${\cal
C}^{\infty}$ it is known to be exponential $\mu \propto \exp (-\alpha T)$.
