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