Lecture+11+-+Quantum+mechanics

Lecture+11+-+Quantum+mechanics - Schrdinger Equation for...

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Schrödinger Equation for Hydrogen Atom H-atom has one electron and one proton. Proton fixed and electron can move in three dimensions (x, y, z). Potential energy (V) given by coulombic interaction: 0 ) ( 8 2 2 2 2 2 2 2 2 = + + + ψ πψ ψψ V E h m z y x e (x) E (x) V dx (x) Ψ d m 8 π h n 2 n 2 2 2 n n Ψ = Ψ + r e Z V 0 2 2 4 πε = e - + r ) ( ) ( ˆ x E x H n n n Ψ = Ψ n=1,2,3 …
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Energy of H-atom is Quantized Principal quantum number n determines total energy of the electron: Same as Bohr Model for the case Z = 1. n = 1, 2, 3 … A e m h a e 529 . 0 2 2 0 0 = = π ε Bohr radius × = Δ 2 2 18 1 1 10 179 . 2 i f n n J E e - + 3 2 1 2 2 0 2 4 2 1 8 n h e m Z E e n = × × = = 2 18 2 0 0 2 1 10 179 . 2 1 8 n J n a e E n πε hcR H hcR H
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Wavefunction in Polar Coordinates [ ] () 0 8 sin 1 ) / ( sin sin 1 2 2 2 2 2 2 2 2 2 = + + + + + ψ π φ θ θψ ψψ V E h m r r r r r e e - + r Φ(φ) Θ(θ) R(r) = radial wavefunction ) ( ) ( ) ( ) , , ( ' ) , , ( Φ × Θ × = Ψ = Ψ r R r z y x r e Z V 0 2 2 4 πε = Φ Θ = φφ θθ 2 0 2 0 2 0 2 ) ( ) ( ) ( d d dr r R prob r
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Wavefunctions for H-atom [ ] 0 4 8 sin 1 ) / ( sin sin 1 2 0 2 2 2 2 2 2 2 2 2 = + + + + + ψ πε π φ θ θψ ψψ r e E h m r r r r r e Ψ n m (r, θ , φ ) described by three quantum numbers (n, , m ): - Principal quantum number, n ( e- distance from nucleus and energy ) n = 1, 2, 3, … - Angular momentum quantum number, ( shape of wavefunction ) = 0, 1, 2, …, (n-1) - Magnetic quantum number, m ( orientation of wavefunction in space ) m = - , …, -1, 0, 1, …, + ) ( ) ( ) ( ) , , ( Φ × Θ × = Ψ r R r l nlm
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Hydrogen Atom Quantum Numbers n = 1, 2, 3, 4 Analogous to Bohr orbits (determines size of orbital) Shell of e- determined by principal quantum number, n: Subshell of e- determined by angular momentum ( = 0, 1, … n-1): Value of 0 1 2 3 4 Name of subshell s p d f g Magnetic quantum number (m
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This note was uploaded on 09/16/2011 for the course CHEM 107B taught by Professor Jamesames during the Summer '09 term at UC Davis.

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Lecture+11+-+Quantum+mechanics - Schrdinger Equation for...

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