Structure of the Xp Matrices in the Simple Harmonic Oscillator Representation
John E. Kilpatrick, Ronald L. Sass · The Journal of Chemical Physics · 1965
The m, n element of Xp in the one-dimensional simple harmonic oscillator representation is shown to be in the form 〈m | xp | n〉=βp|2[(N−r+1)(N−r+3)(N−r+5)···(N+r−1)]12(p12(p−r))2−p/2Frp(N),where N=m+n+1, r=m—n, β=ℏ/2μω and Frp(N) is a simple polynomial in N, purely even or odd and with small, positive, integral coefficients. A number of simple recursion relations are developed for the F polynomials.