Lecture24

Lecture24 - 1 1 PHYS 360 Quantum Mechanics Wed Mar 9 2011...

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Unformatted text preview: 3/9/11 1 1 PHYS 360 Quantum Mechanics Wed Mar 9, 2011 Lecture 24: Finally, the hydrogen atom. Did Bohr get anything right? The Hydrogen Atom First, let’s consider the Bohr model. This was before quantum mechanics, but it did assume quanpzapon. 1. Classical electron orbits, circular, but with quanpzed angular momentum: mvr = n 2. Transipons between orbits leads to emission of a ¡hoton: E γ = ω = E n − E m Note: there are NO wave funcpons. Then insert Coulomb force into Newton’s 2 nd law: F = ma = mv 2 r = Ze 2 r 2 2 Note: much of Bohr’s original argument was based on the Corres¡ondence Princi¡le. Also note: A¢er the de Broglie hy¡othesis was made (that ¡=h/λ), it was found that quanpzed angular momentum meant an integer number of wave lengths ¡er orbit. 3 The Bohr Model Predicpons: Ground state binding energy E = 1 2 mc 2 ( Z α ) 2 = 13.6 eV Bohr radius a o = α mc 0.529 × 10 − 10 m This was sufficient to ¡rovide good agreement with all of the known s¡ectrosco¡ic data at that pme. α ≡ e 2 c 1 137 fine structure constant ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 4 Back to QM in three dimensions….. 5 6 i ∂Ψ ( r , t ) ∂ t = − 2 2 m ∇ 2 Ψ ( r , t ) + V ( r , t ) Ψ ( r , t ) Ψ n r , t ( ) = ψ n ( r ) e − iE n t / − 2 2 m ∇ 2 ψ + V ψ = E ψ ψ ( r , θ , φ ) = R ( r ) Y ( θ , φ ) = R ( r ) Θ ( θ ) Φ ( φ ) 1 R d dr r 2 dR dr ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − 2 mr 2 2 V ( r ) − E ⎡ ⎣ ⎤ ⎦ = l l + 1 ( ) The Radial Equapon: 1 Θ sin θ d θ sin...
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Lecture24 - 1 1 PHYS 360 Quantum Mechanics Wed Mar 9 2011...

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