On the energy spectra of one dimensional and harmonic oscillators

Abstract

In this paper we present explicit and simple analytical formulae for the energy eigenvalues E_n(λ) of one-dimesional anharmonic oscillators characterized by the potentials ½m_ω²x²+λx^2α with α = 2, 3 and 4. A simple intuitive criterion supplemented by the requirment of correct asymptotic behavour, has been employed in arriving at the formulae. Our energy values over a wide range of n and λ are in good agreement with the numerical values computed by earlier workers through very elaborate techniques. To our knowledge this is the first time that formulae of such wide validity have been given. The results of pure power oscillators are rivally obtained by going over to the ω→0 limet. Approximate analytic expressions for th low order even moment of x are also given.