Exact solution of the compressed hydrogen atom

Abstract

The exact solution to the problem of a hydrogen atom confined in a spherical well (CHA) is discussed; the standard results for the unconfined hydrogen atom (UHA) are recovered as the sphere size becomes large compared to the Bohr radius. The solutions are characterized by a set of three quantum numbers N (= 1, 2, 3,...), L (= 0, 1, 2,...), and M (= - L, - L + 1,..., L - 1, L), and the energy eigenvalues, in contrast to the situation in the UHA, depend on both N and L. All members of a given family n = N + L, however, evolve asymptotically toward the same energy level in the large-sphere limit, recovering the typical n2 degeneracy of the UHA. Besides numerically exact solutions for arbitrary sphere sizes, rigorous analytical approximations are provided for the physically relevant strong- and weak-confinement regimes. A conjecture concerning the ordering of the energy levels is rigorously confirmed. The validity of the virial theorem, Kato's cusp condition, and the role played by the density as an alternative basic variable for the case of the CHA are discussed. © 2013 American Association of Physics Teachers.