Quantum theory of anharmonic oscillators. I. Energy levels of oscillators with positive quartic anharmonicity

Abstract

This is an investigation of the energy levels of an anharmonic oscillator characterized by the potential (1/2) x²+λx⁴. Two regions of λ and n are distinguishable (n being the quantum number of the energy level) one in which the harmonic oscillator levels E_n=n+1/2 are only slightly distorted and the other in which the purely quartic oscillator form E_n?cλ^1/3(n+1/2)^4/3 (c being a constant) is only slightly distorted. Rapidly converging algorithms have been developed, using the Bargmann representation, from which energy levels in any (λ,n) (with λ≳0) regime can easily be computed. Simple formulas are also derived which give excellent approximations to the energy levels in various (λ,n) regimes.