Closed-Form Hydrogenic RadialrkMatrix Elements and the Factorization Method

It is shown that the factorization method, when introducing accelerated ladder operators or accelerated ladder matrices, leads to a formula in closed form for the general off-diagonal ($n\ne n^{\prime }$) hydrogenic $r^k$ matrix elements. This explicit expression, which involves only factorials and binomial coefficients, is directly reducible to any particular case. The well-known selection rules follow from the formula. The method seems appropriate to other cases of factorizable equations. Its extension to the Dirac-Coulomb, generalized Kepler problems, and the discrete-continuous case is considered.