Lecture18.pptx

Lecture18.pptx - 2/23/11 PHYS360QuantumMechanics...

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2/23/11 1 PHYS 360 Quantum Mechanics Wed Feb 23, 2011 Lecture 18: What if the square well isn't in±nitely deep? (Bound states and sca²ering states) HW 5 due on Monday: Ch 2, #23, 30, 34, 35 Recap: delta funcUon potenUal 2 V x ( ) = αδ x ( ) h 2 2 m d 2 ψ dx 2 x ( ) = E d 2 dx 2 = 2 mE h 2 = κ 2 2 mE h Consider the delta funcUon potenUal well: SUck into Schrödinger Eqn: The usual rearrangement, for x away from the origin: …with: At ±rst we consider only bound states, where E<0. 3 1. is always continuous; 2. d dx is continuous except at points where the potential is infinite x ( ) = Be x , x 0 ( ) , Be −κ x , x 0 ( ) ; 4 2 2 m d 2 dx 2 + V x ( ) x ( ) dx = E x ( ) −∈ + −∈ + −∈ + dx Δ d dx lim ∈→ 0 d dx + d dx −∈ = 2 m 2 lim ∈→ 0 V x ( ) x ( ) dx −∈ + We need to ±nd the energy eigenvalue E and the normalizaUon constant B . Integrate the Schrödinger equaUon across the “disconUnuity”: (What happened to E term?) 5 E = 2 2 2 m = m α 2 2 2 B = = m x ( ) = m e m x 2 ; E = m 2 2 h 2 Normalize the wave funcUon: Finally: The delta funcUon potenUal well has exactly one bound state.
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Lecture18.pptx - 2/23/11 PHYS360QuantumMechanics...

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