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arXiv:quant-ph/9412010 · Journal of Physics A Mathematical and General

Quantum averaging. I. Poincare-von Zeipel is Rayleigh Schrodinger

W Scherer

Published Dec 21, 1994• 29 citations
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Abstract

An exact analogue of the method of averaging in classical mechanics is constructed for self-adjoint operators. It is shown to be completely equivalent to the usual Rayleigh Schrodinger perturbation theory but gives the sums over intermediate states in closed-form expressions. The anharmonic oscillator and the Henon-Heiles system are treated as examples to illustrate the quantum averaging method.

DOI
10.1088/0305-4470/27/24/028

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An exact analogue of the method of averaging in classical mechanics is constructed for self-adjoint operators. It is shown to be completely equivalent to the usual Rayleigh Schrodinger perturbation theory but gives the sums over intermediate states in closed-form expressions. The anharmonic oscillator and the Henon-Heiles system are treated as examples to illustrate the quantum averaging method.
The authors develop a simple alternative approach to perturbation theory in one-dimensional nonrelativistic quantum mechanics. The formulae for the energy shifts and wavefunctions do not involve cumbersome sums over intermediate states which appear in the usual Rayleigh-Schrodinger expansion. Unlike standard logarithmic perturbation theory, the approach does not utilize taking the logarithm of the wavefunction, and is therefore applicable in the same form to both the ground state and excited bound states.
The accuracy of the renormalized perturbation series for anharmonic oscillators and other simple quantum-mechanical models increases with the quantum number when the adjustable parameter is determined by a simple variational criterion. In the semiclassical limit of quantum mechanics such an expansion converges towards the result of the Jeffreys–Wentzel–Kramers–Brillouin method to all orders. The hypervirial and Hellmann–Feynman theorems facilitate the derivation of the semiclassical series from Rayleigh–Schrödinger perturbation theory.