The theory of adiabatic invariants has a long history and importantapplications in physics but is rarely rigorous. Here we treat exactly thegeneral time-dependent 1-D harmonic oscillator, $\ddot{q} + \omega^2(t) q=0$which cannot be solved in general. We follow the time-evolution of an initialensemble of phase points with sharply defined energy $E_0$ and calculaterigorously the distribution of energy $E_1$ after time $T$, and all itsmoments, especially its average value $\bar{E_1}$ and variance $\mu^2$. Usingour exact WKB-theory to all orders we get the exact result for the leadingasymptotic behaviour of $\mu^2$.
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机译：绝热不变量理论在物理学中有着悠久的历史和重要的应用，但很少严格。在这里，我们精确地对待一般时间相关的一维谐波振荡器$ \ ddot {q} + \ omega ^ 2（t）q = 0 $，这通常是无法解决的。我们遵循具有清晰定义的能量$ E_0 $的相点初始集合的时间演化，并严格计算时间$ T $之后的能量$ E_1 $的分布及其所有矩，尤其是其平均值$ \ bar {E_1} $和方差$ \ mu ^ 2 $。对所有阶使用精确的WKB理论，我们得到$ \ mu ^ 2 $的前渐近行为的精确结果。
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