Lect28_HydrogenI - Review of Center-of-mass coordinates:...

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Lecture 28: The Hydrogen Atom PHY851 Quantum Mechanics I Fall, 2008 M.G. Moore Review of Center-of-mass coordinates: For the problem of two interacting particles, we have: Center-of-mass and relative coordinates defined by: Hamiltonian transforms to: The general form of the solutions is: Where: ( ) 2 1 2 2 2 1 2 1 2 2 R R V m P m P H r r ! + + = 2 1 2 2 1 1 m m R m R m R CM + + = r r r 2 1 R R R r r r ! = 2 1 P P P CM r r r + = 2 1 2 1 1 2 m m P m P m P + ! = r ( ) R V P M P H CM + + = μ 2 2 2 2 2 1 2 1 2 1 m m m m m m M + = + = ) ( ) ( 2 ) 1 ( 2 ) ( , 2 2 2 2 2 , , r R r V mr r m r R E n n n l l l l l h h ! ! " # + + + $ $ % & ( = ) , ( ) ( , , , , , 1 ! " m n Y r R r m n r l l l # =
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Example: Hydrogen Atom The Hamiltonian for a system consisting of an electron and a proton is: In COM coordinates, the Hamiltonian for the relative motion is: The radial wave equation is: With the dimensionless variables: The radial equation becomes: p e p p e e R R e m P m P H r r ! ! + = 0 2 2 2 4 2 2 "# R e P H 0 2 2 4 2 !" μ # = ) ( 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 ! ! " # $ + + % % & ( ( $ = )* ! " n r = 2 2 2 n n E h " = ) ( ) ( , n n R u l = 2 0 2 4 2 h # $ n n e = u u n ! ! " # $ $ % & + + = ( ( ) 2 ) 1 ( 1 l l Series Solution We skipped the series solution approach for the
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This note was uploaded on 10/25/2010 for the course PHYSICS PHYS 851 taught by Professor Michaelmoore during the Fall '08 term at Michigan State University.

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Lect28_HydrogenI - Review of Center-of-mass coordinates:...

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