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

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.Your membership status does not allow you to participate in discussion or see all comments.

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