Lecture18 - WedFeb23,2011 Lecture18 'tinnitelydeep( Recap:...

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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 infinitely deep? (Bound states and scaMering 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 αδ x ( ) ψ = E ψ d 2 ψ dx 2 = 2 mE h 2 ψ = κ 2 ψ κ 2 mE h Consider the delta funcUon potenUal well: SUck into Schrödinger Eqn: The usual rearrangement, for x away from the origin: …with: At first we consider only bound states, where E<0. 3 1. ψ is always continuous; 2. d ψ dx is continuous except at points where the potential is infinite ψ x ( ) = Be κ x , x 0 ( ) , Be κ x , x 0 ( ) ; 4 2 2 m d 2 ψ dx 2 + V x ( ) ψ x ( ) dx = E ψ x ( ) −∈ + −∈ + −∈ + dx Δ d ψ dx lim ∈→ 0 d ψ dx + d ψ dx −∈ = 2 m 2 lim ∈→ 0 V x ( ) ψ x ( ) dx −∈ + We need to find the energy eigenvalue E and the normalizaUon constant B . Integrate the Schrödinger equaUon across the “disconUnuity”: (What happened to E term?) 5 E = 2 κ 2 2 m = m α 2 2 2 B = κ = m α ψ x ( ) = m α e m α x 2 ; E = m α 2 2 h 2
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