Closed-form expressions of matrix elements and eigenfunctions from ladder-operator considerations

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Abstract
Within the Schrödinger‐Infeld‐Hull factorization scheme, it is shown that, by suitable transformations, the ``accelerated'' or ``ν‐step'' ladder operator can always be brought to a simple canonical form, i.e., the νth derivative operation. Thus, one obtains a closed form expression of the eigenfunctions involving a Rodrigues' formula. The necessary and sufficient condition that this Rodrigues' formula generates classical orthogonal polynomials is found to be equivalent to the factorizability condition. Consequently, a closed form expression of any matrix element (diagonal or off‐diagonal) on the basis of the eigenfunctions of any factorizable equation is easily derived from the calculation of one unique particular integral. In most cases, this last integral is known analytically. The Kepler problem is reinvestigated as an example. As a concluding remark, further applications of the method are considered.