Matrix elements of an arbitrary potential in the harmonic oscillator basis

Abstract

We derive a formula for the matrix elements of any potential V( x) in the one-dimension harmonic oscillator representation. It is f nm = n!m! π2 n+m 1 2 ∑ k=0 n 2 K K!(n−K)!(m−K)! T n+m−2K· For an N × N matrix only 2 N − 1 T r 's need to be calculated. Each T r is proportional to the top row matrix element f 0, r and several ways are given to calculate them. This method of evaluating the elements is particularly useful if the potential is given numerically. We also give a useful four-term recursion relation between the matrix elements. Examples worked out are the potentials x j and a shifted Gaussian e −( x− a) 2 .