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Lecture 26 typed - Lecture 26 Further Into the Hydrogen...

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Lecture 26: Further Into the Hydrogen Atom Additional reading AF ch. 3.9-3.14 Last lecture we separated the arrived at a form for the internal wavefunction of the hydrogen atom. 2 2 2 2 2 2 2 2 2 2 2 0 2 ( ) ( ) upon which we elaborate in shperical coordinates 2 4 2 1 1 ( , , ) sin ( , , ) 4 sin sin 2 ( , , r n o Ze r E r r r Ze r r r r r r r E r ψ ψ μ πε μ ψ θ ϕ θ ψ θ ϕ πε θ θ θ θ φ μ ψ θ = + + = arrowrightnosp arrowrightnosp arrowrightnosp ) where the first bracketed term on the left is purely radial and the second is purely angular. thus we can assert that ( , , ) ( ) ( , ) which includes our old friends the spherical harmonics! lm r R r Y ϕ ψ θ ϕ θ φ = 2 2 2 2 2 2 0 2 2 2 To show this more robustly, 2 ˆ ( ) ( , ) ( ) ( , ) ( ) 4 1 ˆ [ radial terms ] ( ) ( , ) ( ) ( , ) set both sides to a constant - for two equa lm lm lm lm r Ze r E R r Y L R r Y R r r r r
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