Matrix Elements of General Potentials in the Harmonic-Oscillator Representation

Abstract

The m, n matrix element of an arbitrary potential function V(q) in the one-dimensional harmonic-oscillator representation is shown to be given by 〈m | V(q) | n〉= ∑ r=0mα(m!n!)12r!(m−r)!(n−r)! ∫ −∞∞dyg(αy)e−12y2(iy)m+n−2r,where α≡(2ω/ℏ)½, and g(αy) is the Fourier transform of V(q). This formula is specialized to the cases where V(q) is given by qj, qje−½γq2, ejαzq, and q−1 sin(αλq), where j is a nonnegative integer and γ, z, and λ are real parameters. Results are compared, where possible, with previous work.