Lecture 17 - Quantum mechanics

# Lecture 17 - Quantum mechanics - Problem Set 2 due at end...

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Unformatted text preview: Problem Set 2 due at end of class on Fri Feb 18 th . Midterm 1 key on website. 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: ) ( 8 2 2 2 2 2 2 2 2 = − + ∂ ∂ + ∂ ∂ + ∂ ∂ ψ π ψ ψ ψ 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 2 2 4 πε − = e- + r ) ( ) ( ˆ x E x H n n n Ψ = Ψ n=1,2,3 … 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 . 2 2 = = 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 2 4 2 1 8 n h e m Z E e n ε − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ × × − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = − 2 18 2 2 1 10 179 . 2 1 8 n J n a e E n πε hcR H 1 2 3 n = 4 n = 1 n = 2 n = 3 n = 4 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ × × − = − 2 18 1 10 179 . 2 n J E n Quantized Energy Levels of H-atom ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − × − = Δ − 2 2 18 1 1 10 179 . 2 i f n n E Absorption Emission ( Δ E = h ν ) 4 3 2 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = 2 1 n hcR E H n Energy Levels of Hydrogen Emission ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − × − = Δ − 2 2 18 1 1 10 179 . 2 i f n n E nm E hc 120 1 2 = Δ = → λ nm E hc 486 2 4 = Δ = → λ Wavefunction in Polar Coordinates [ ] ( ) 8 sin 1 ) / ( sin sin 1 2 2 2 2 2 2 2 2 2 = + + ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ ψ π φ ψ θ θ θ ψ θ θ ψ ψ V E h m r r r r r e e- + r...
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Lecture 17 - Quantum mechanics - Problem Set 2 due at end...

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