Quantum theory of anharmonic oscillators. II. Energy levels of oscillators with <i>x</i>2α anharmonicity

Abstract

This is an investigation of the energy levels of anharmonic oscillators characterized by the potentials (1/2) x2+λx2α with α=2,3,⋅⋅⋅ and λ≳0. Two regimes of (λ,n) space are distinguishable: In one, the energy levels differ only slightly from the harmonic ones En=n+1/2 and in the other they differ only slightly from the purely anharmonic oscillators with En?Cαλ1/(1+α) (n+1/2)2α/(1+α), C being a constant which depends on α. The magnitude of the combination η≡λ (n+1/2)α−1 determines the regime. If this combination is ≪1, one is in the harmonic regime, and if it is ≫1, one is in the anharmonic regime. As n→∞ the ’’boundary layer’’ between the two regimes narrows. The small parameter of a perturbation theory should thus be η rather than λ. Several rapidly convergent algorithms have been developed for the calculation of the energy levels of our anharmonic oscillators, and energy level tables and graphs are presented.