Radial matrix elements of rn for twofold and threefold isotropic simple harmonic oscillators

A generating function for associated Laguerre polynomials and Louck's recursion relations have been employed independently to formulate radial matrix elements of rⁿ for g-fold (g = 2, 3) isotropic simple harmonic oscillators in terms of quantum numbers v and l. This paper gives all relevant matrix elements (vl|rⁿ|v′l′) of rⁿ for n = 1,2, …, 7 and the diagonal element of r⁸. Introduction of the quantities, $\text{V =}\text{1}\text{2}\text{(v + v′ + g)}$ and $\text{L =}\text{1}\text{2}\text{(l + l′ + g ? 2)}$, simplifies the expressions and reveals symmetries and patterns occurring as n increases. Results are compared with those for the one-fold oscillator and sample computer routines have been prepared for numerical evaluation of the matrix elements for g = 1, 2, 3.