lecture21to22 - 5.61 Fall 2007 Lecture#21 page 1 HYDROGEN...

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5.61 Fall 2007 Lecture #21 page 1 HYDROGEN ATOM Schrodinger equation in 3D spherical polar coordinates: ! 2 1 ∂ ⎛ 2 ∂ ⎞ 1 ∂ ⎛ ∂ ⎞ 1 2 2 µ r 2 r r r + r 2 sin θ θ sin θ θ + r 2 sin 2 θ φ 2 ψ ( r , θ , φ ) + U ( r , θ , φ ) ψ ( r , θ , φ ) = E ψ ( r , θ , φ ) Ze 2 with Coulomb potential U ( r ) = 4 πε 0 r Rewrite as 2 ! 2 ∂ ⎛ ∂ ⎞ 2 U r r , θ , φ ) + L ˆ 2 ψ r , θ , φ ) = 0 + 2 µ r ( ) E ψ ( ( r r r function of r only function of θ , φ only r is separable ψ is separable Angular momentum: solutions are spherical harmonic wavefunctions m ψ r , θ , φ ) = R r θ , φ ( ( ) Y ( ) l m m with L ˆ 2 Y ( θ , φ ) = ! 2 l l ( + 1 ) Y ( θ , φ ) l = 0,1,2,... l l Radial equation for the H atom: dR r ! 2 l l + 1 ! 2 d ( ) + ( ) ( ) E ( ) = 0 2 r + U r R r 2 µ r 2 dr dr 2 µ r 2 Solutions R r ( ) are the H atom radial wavefunctions Simplest case: l = 0 yields solution ( ) = 2 a Z 0 3 2 e Zr a 0 R r exponential decay away from nucleus
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5.61 Fall 2007 Lecture #21 page 2 with E = Z 2 e 2 8 πε 0 a 0 lowest energy eigenvalue π µ e 2 Bohr radius a 0 ε 0 h 2 General case: solutions are products of (exponential) x (polynomial) Energy eigenvalues: E = Z 2 e 2 = Z 2 µ e 4 n = 1,2,3,... 8 πε 0 a 0 n 2 8 ε 0 2 h 2 n 2 Radial eigenfunctions: 1 2 l + 3 2 R nl ( ) = ( n l 1 ) ! 3 na 2 Z 0 l Z r na 0 L 2 l + 1 2 na Zr 0 r r e n + l 2 n ( n + l ) !
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