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The radial part of the wave function of an electron in a Coulomb potential isthe product of a Laguerre polynomial and an exponential with the variablescaled by a factor depending on the degree. This note presents an elementaryproof of the orthogonality of wave functions with differing energy levels. Itis also shown that this is the only other natural orthogonality for Laguerrepolynomials. By expanding in terms of the usual Laguerre polynomial basis ananalogous strange orthogonality is obtained for Meixner polynomials.

A LAGUERRE POLYNOMIAL ORTHOGONALITY AND THE HYDROGEN ATOM