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Infinite-Rydberg Limit of the Hydrogen Atom: The Lowest-Energy Unbound States

Michael H. Brill
Year: 2011 Pages: 2
The radial wavefunctions of the hydrogen atom have an interesting mathematical behavior in the limit of infinite n, which is the infinite Rydberg limit. Finding the limiting wavefunctions corresponds to solving the radial Schroedinger equation for E = 0. Frobenius expansion gives two solutions, each of which is a convergent expansion, but neither is square-integrable. For bound states, the second Frobenius solution is discarded because a relation between the solutions at r = 0 is violated and the square-integrable solution survives. We can\'t rule out either one of the E = 0 solutions solutions on the same grounds, which highlights the fact that half the mathematical solutions for the energies E < 0 have been artificially wiped away. We have lived without a probability interpretation for all the unbound-state wave functions, so I conjecture here that there may be physical significance to the non-integrable wave functions at E < 0 that are ignored in textbooks.