Two-step approach to one-dimensional anharmonic oscillators

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Abstract
We propose a two-step approach to one-dimensional anharmonic oscillators. A generalized coherent-state ansatz is introduced for the first step. A theorem of Wick's ordering and a Bogoliubov transformation are used to simplify the derivation. This is shown to be equivalent to the Hartree approximation. In the second step, standard diagonalization is used for the transformed Hamiltonian. The method yields a clear physical picture and is capable of producing accurately all the low-lying energy levels in a single diagonalization. Asymptotic expansions for the energy levels are easily obtained. The connection with a pure quartic oscillator is pointed out and as a by-product no calculation is necessary in that model. We also include cubic couplings and apply the method successfully for the two-well oscillator.

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