Rayleigh-Schrödinger perturbation theory for the anharmonic oscillator

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Abstract
A Rayleigh-Schrödinger perturbation theory is developed for the anharmonic oscillator with a quartic anharmonicity term. The theory is developed in two steps. First a new ground state is chosen variationally and then a basis is built on this ground state by defining a new set of creation and annihilation operators. The Hamiltonian is written down in occupation-number space in terms of these new operators. Then the perturbation theory is developed by putting all diagonal terms as the unperturbed Hamiltonian. The perturbation series is shown to be convergent. The theory allows for the correct limit in both large and small anharmonicity.

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