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HydrogenAtom - Physics 4143 Quantum Mechanics II Brian...

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Physics 4143: Quantum Mechanics II Brian Kennedy Hydrogen atom bound states via the hypergeometric equation, Spring 2011 For those who would like to test their understanding of the H-atom bound state calculation, these notes/exercises take you through a slightly different argument to that given in Griffiths’s book. The MATHEMATICA notebook given in the T-square Resources tool illustrates the solution. In Schr¨ odinger’s equation for the relative radial motion of an electron in a Coulomb potential we set the relative separation r = αρ , where ρ is dimensionless separation and α an, as yet, unspecified length scale : ( - ~ 2 2 µα 2 d 2 2 - Ze 2 α 1 ρ + l ( l + 1) ~ 2 2 µα 2 1 ρ 2 ) u = Eu If we are interested in bound states E = -| E | < 0, and we futher define | E | = ~ 2 / (8 µα 2 ) and λ Ze 2 / (4 α | E | ), the Schr¨ odinger equation reduces to ( d 2 2 + λ ρ - l ( l + 1) ρ 2 - 1 4 ) u = 0 . The required form of the solution near ρ = 0 and ρ = , suggests that the substitution u = ρ l +1 e - ρ/ 2 f ( ρ
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