11_Hatom_new

Electron spin eigenvalues 1 12 12 spin quantum number

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Unformatted text preview: ) 1 1 Z 2 n 2 H 1s <r>=1.5 l 3 / 2 He+ 1s <r>=0.75 Z2 E 2 2n atomic units Li2+ 1s <r>=0.5 21 continuum 9/30/2013 n= 3p 3s 3,1,0 3,1,1 3,1,-1 3,0,0 2p 2,0,0 2,1,0 2,1,1 2,1,-1 Energy 2s 3d 3,2,0 3,2,1 3,2,-1 3,2,2 3,2,-2 1s 1,0,0 Bohr Model n, , m Schrödinger Model - Nonrelativistic continuum 3s 3,0,0 3p 3d 3,1,0 3,1,1 3,1,-1 3,2,0 3,2,1 3,2,-1 3,2,2 3,2,-2 2p 2,0,0 2,1,0 2,1,1 2,1,-1 Energy scale of changes exaggerated 2s j 1/ 2 going beyond our work → 1s 1,0,0 Bohr Model n, , m scale of changes exaggerated Schrödinger - Nonrelativistic Dirac, Fine Structure, Relativistic Lamb Shift, QED Hyperfine Structure, Proton Spin 22 9/30/2013 Plot a 50% Contour for any H-atom Orbital derviative function only works up to 4,3,2 it won't do 4,3,3 n 4 l 3 m 0 prinicipal orb. angular momentum projection of l pick the quantum numbers define a general wavefunction st define associated Laguerre function L( s t x) k 0 2 k 1 k ( s ) ( 1) x ( s t k) ( t k) k l radial wavefunction R( r n l) 3 2 ( n l 1) 2 m r n 2 n ( ( n l) ) 3 n l r e L n l 2 l 1 2 r n m define associated Legendre polynomial P( l m x) ( 1) l 2 1x 2 l angular wavefunction, spherical harmonic total wavefunction Y l m ( 2 l 1) ( l m ) 4 ( l m ) 2 d l m l m x2 1 l dx P l m cos exp im n l m r R( r n l) Y l m 23 9/30/2013 plot the wavefunction find rmax and thetamax numerically i i 40 1000 1i i 1000 r find rmax associated with 50% contour i 2 Pr R( r n l) R( r n l) r dr i 0 for i 0 1000 imax r i 0 1000 break if Pr 0.5 imax 434 i rmax r imax rmax 17.36 i YsY Y 1i 0 l m Y 1i 0 l m find theta at max imax i YsYmax max( YsY) YsYmax 0.557 for i 0 1000 break if YsY i YsYmax imax 0 max 1imax max 0 i 2 rmax Y l m sin cos 2 Y max 0 l m 2 Y l m sin sin G rmax 2 Y max 0 l m Y l m 2 cos rmax 2 Y max 0 l m This function takes values of and and maps them into x, y, and z components (stacked in the column vector). The radial value is fixed at the max of the radial probability distribution and the (Y*Y)/(Y*maxYmax) scales to one at the angular max. p CreateMesh ( G 120 120) p 24 9/30/2013 A hydrogen atom walks into a bar and orders a drink. The bartender says, "Gee you look a little down. Is something the matter?" The hydrogen atom says, "Yes, I've lost an electron." The bartender asks, "Are you sure?" "I'm positive." IP 25...
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