Transition from classical mechanics to quantum mechanics: $x^4$ perturbed harmonic oscillator

The detailed transition from classical mechanics to quantum mechanics, particularly in regard to the question of quantum corrections to the semiclassical quantization, can be completely understood for the $x^4$ perturbed harmonic oscillator. This is a consequence of five facts: (i) The quantum perturbation series for the energy can be solved order by order in closed form. (ii) The quantum series rearranges directly into the classical canonical perturbation series plus quantum corrections proportional to successively higher powers of ${ħ}^2$. (iii) The classical series and the subseries for the quantum corrections converge (subseries by subseries) to the terms of the Jeffreys-Wentzel-Kramers-Brillouin (JWKB) expansion for the energy (to all orders of ħ). (iv) The (all-order) JWKB expansion, conversely, yields the quantum perturbation series and consequently uniquely determines the exact eigenvalue. (v) The ratio of the quantum energy to the classical energy at action (n+1/2)ħ differs from 1 asymptotically by 1/[9π(n${+1/2)}^2$], independent of both ħ and the perturbation coupling constant, as the harmonic-oscillator quantum number n approaches ∞. An interesting additional feature is the mechanism by which the convergence of the classical perturbation series and of the series for the quantum corrections gets turned into the divergence of the quantum perturbation series.