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Lect29_HydrogenII - Review of QM Hydrogen The relative...

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Lecture 29: Hydrogen II PHY851 Quantum Mechanics I Fall, 2008 M.G. Moore Review of QM Hydrogen The relative motion of the electron-proton pair is governed by the Hamiltonian: With some work, we were able to find the eigenstates: m n E m n H n , , , , l l = l K l l K l K , , 1 , 1 , , 1 , 0 , 3 , 2 , 1 + ! ! = ! = = m n n 2 2 0 2 1 2 n a E n μ h ! = J 10 18 . 2 eV 6 . 13 2 18 2 0 2 1 ! " = ! = ! = a E μ h 2 n d n = R e P H 0 2 2 4 2 !" μ # = m n m n L , , ) 1 ( , , 2 2 l l l h l + = m n m m n L z , , , , l h l = ( ) ( ) ! " ! " , 2 2 2 ! 1 )! 1 ( 2 , , , , 0 ) 1 2 ( 1 0 / 3 0 0 m n n a r Y n a r L n a r e n n n n a m n r l l l l l l # $ $ % & ( ) $ $ % & ( ) * * $ $ % & ( ) + = + * * *
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Energy Level Diagram By taking spin into account (two spin states per orbital), and using the Pauli principle, the degeneracies of these levels explain much of the structure of the periodic table: 2, 8, 18, 32,…. E 0 n= 1 n= 2 n= 3 n= 4 l l =0 l =1 l =2 l =3 1 s 2 s 2 p 3 s 4 s 3 p 4 p 3 d 4 d 4 f (7) (5) (3) (1) (1) (4) (9) (16) Hydrogen Atom in a Magnetic Field Minimal coupling Hamiltonian: For a static E-field, can just add the scalar potential to the Hamiltonian Most EM fields are not described by a scalar potential, instead they are described by a vector potential To include such EM fields, we can make the transformation: Here q is the charge and A ( R ) is the vector potential The Hamiltonian of an electron then becomes: v P " v P # q
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