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HydrogenAtom - The Hydrogen Atom Michael Fowler Factoring...

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The Hydrogen Atom Michael Fowler 11/22/06 Factoring Out the Center of Mass Motion The hydrogen atom consists of two particles, the proton and the electron, interacting via the Coulomb potential , where as usual ( ) 2 1 2 / V r r e r = G G 1 2 r r r = G G . Writing the masses of the two particles as Schrödinger’s equation for the atom is: 1 , m m 2 ( ) ( ) 2 2 2 2 2 1 2 1 2 1 2 1 2 , , 2 2 e r r E r r m m r ψ ψ ∇ − ∇ − = G G = = . G G G G But are not the most natural position variables for describing this system: since the potential depends only on the relative position, a better choice is 1 2 , r r G G , r R G G defined by: 1 1 2 2 1 2 1 2 , m r m r r r r R m m + = = + G G G G G G so R G is the center of mass of the system. It is convenient at the same time to denote the total mass by 1 2 , M m m = + and the reduced mass by 1 2 1 2 . m m m m m = + Transforming in straightforward fashion to the variables , r R G G Schrödinger’s equation becomes ( ) ( ) 2 2 2 2 2 , , 2 2 R r e . R r E R r M m r ψ ψ ∇ − = G G G G = = G G Writing the wave function ( ) ( ) ( ) , R r R r ψ ψ = Ψ G G G G we can split the equation into two: ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 R R r r R E R M V r r E r m ψ ψ Ψ = Ψ ∇ + = G G G = G = G G G
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