Lect30_Hydrogen - Lecture 30: The Hydrogen Atom Phy851 Fall...

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Lecture 30: The Hydrogen Atom Phy851 Fall 2009
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Example 2: Hydrogen Atom The Hamiltonian for a system consisting of an electron and a proton is: In COM and relative coordinates, the Hamiltonian is separable: The energy eigenvalue equations are: p e p p e e R R e m P m P H r r + = 0 2 2 2 4 2 2 πε r CM H H H + = R e P H r 0 2 2 4 2 μ = M P H CM CM 2 2 = ( ) ) ( ) ( ) ( ) ( R r C CM r CM R r C CM E E E E E E H + = ) ( ) ( ) ( C CM CM C CM C CM E E E H = ) ( ) ( ) ( R r r R r R r E E E H =
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Central Potential Ansatz For a spherically symmetric potential, we can choose to form simultaneous eigenstates of H r , L 2 , and L z : Separate the radial and angular degrees of freedom: The Radial wave equation is then: ) ( 1 4 2 ) 1 ( 2 ) ( , 0 2 2 2 2 2 2 , r R r e r r r R E n n n l l l l h h + + = πε μ ) ( ) ( ) ( , , , , Ω = m n m n r R l l l ) ( , , 1 ) ( r R r n r n r l l = ) , ( , , ) ( φ θ m Y m l l = Ω ) ( ) ( , , R R r m n E l = ) ( ) ( , , , , R n R r m n E m n H l l = ( ) ) ( 2 ) ( 2 , , 1 , , R R m n m n L l l l h l + = ) ( ) ( , , , , R R z m n m m n L l h l =
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Natural Units Introduce dimensionless units:
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Lect30_Hydrogen - Lecture 30: The Hydrogen Atom Phy851 Fall...

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