Hypervirial theorem and ladder operators. Recurrence relations for harmonic oscillator integrals

Abstract

Abstract A second‐quantization formalism combined with a hypervirial theorem is used to derive new recurrence relations for one‐dimensional harmonic oscillator matrix elements. The most general case of 〈 m | f (â, â + )| n 〉 is considered, and the recurrence relations for f (â, â) = X k , exp(−β X ), and exp(− X 2 ) are given as examples. The relations obtained are considerably simpler than those derived by using only the hypervirial theorem; comparatively, the recurrence relations presented here have the advantage of avoiding the use of the quantum mechanical sum‐rules when determining initial matrix elements. The proposed procedure can be used to determine the recurrence relations for other potentials as well as to evaluate the two‐center integrals.