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By Antonio S. De Castro, Centro De Física Computacional and Guaratinguetá Sp Brasil

Abstract

Orthogonality criterion is used to shown in a very simple and general way that anomalous bound-state solutions for the Coulomb potential (hydrino states) do not exist as bona fide solutions of the Schrödinger, Klein-Gordon and Dirac equations. An alleged tightly-bound state of hydrogen with strong singularity of the eigenfunction at the origin (called a hydrino state) has received considerable attention in the literature [1]. The order of magnitude of the atomic size (Bohr radius) as well as the energy of the hydrogen atom in its ground state just derived from the Heisenberg uncertainty principle, even in a relativistic framework, should be enough to disqualify hydrino states. However, in a recent Letter, Dombey [1] rejects the solution of the three-dimensional Klein-Gordon equation, previously derived by Naudts [2], as well as the solution of the two-dimensional Dirac equation, by resorting to a few fair arguments. In addition, Dombey presents a solid argument founded on the Hermiticity of the Hamiltonian for the Klein-Gordon case and a suggestion of similar treatment for the three-dimensional Dirac case. In the wake of Dombey’s suggestion, this Letter presents such a general criterion for banishing hydrino states in the context of the standard quantum mechanics. The time-independent Schrödinger equation and the time-independent Klein-Gordon equation − �2 2M ▽2 ψ + V ψ = Eψ (1) − � 2 c 2 ▽ 2 ψ + M 2 c 4 ψ = (E − V) 2 ψ (2) with spherically symmetric potentials admit eigenfunctions in the form ψ = uk(r) r Ylm (θ, φ) (3) where k denotes the principal quantum number plus other possible quantum numbers, uk is a square-integrable function ( ∫ ∞ 0 dr |uk | 2 = 1) and Y m l are the orthonormalized spherical harmonics ( ∫ dΩY ∗ lmY˜l ˜m = δl˜l δm ˜m), with l = 0, 1, 2,... and m = −l, −l + 1,...,l, in such a way that Heffuk = (Eeff) k uk (4) wit

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