notes03 - 5.73 Lecture #3 3-1 Reading Chapter 1, CTDL,...

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5.73 Lecture #3 3 - 1 Reading Chapter 1, CTDL, pages 9-39, 50-56, 60-85 h 2 E n = n 2 8mL 2 Last time: 1. 1-D infinite box d ψ d 2 ψ ψ n = (2 / L) 1/2 sin(n π x) continuity of ψ (x), dx , dx 2 confinement quantization 2. δ –function well ma 2 E = one bound level 2 h 2 ma 1/2 e ma|x|/ h 2 (what happens to ψ as a increases?) ψ= ± h 2 Why do we know there is only one bound level? What do we know about ψ (p) ? How does this depend on a ? what about <p>? TODAY and WEDNESDAY : 1. motion time dependent Schr. Eq. 2. motion of constant phase point on Ψ (x,t) -- phase velocity 3. motion of | Ψ (x,t)| 2 requires non-sharp E 4. encode Ψ( x,t ) for x 0 , ∆ x , p 0 , p 5. p 0 , p from |g(k)| 6. x 0 , x from stationary phase argument 7. moving, spreading wavepacket | Ψ (x,t)| 2 8. group velocity phase velocity -- see CTDL, pages 28-31 revised 9/4/02 10:35AM
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5.73 Lecture #3 3 - 2 1. Motion time dependent Schr. Eq. i h ∂Ψ = H Ψ TDSE t if V(x) is time independent, then Ψ n (x,t) n (x)e iE n t/ h can use this form of Ψ to satisifies TDSE?± describe time dependence of any non-eigenstate initial preparation: e.g. wavepackets Ψ (,0) = a n ψ n ( x ) superposition of eigenstates x xt Ψ (, ) = a n ψ n ( x ) e i ω n t ω n = E n / h go back to free particle to really see motion of QM systems ψ || () = Ae ikx + Be ikx k x 22 2 E k V 0 = h k = p WHAT ABOUT 2m 2 ) m ARBITRARY ω k = ( E k V 0 h = h k 2 0 ZERO OF E? 2m add a phase factor which expresses the arbitrariness of the zero of energy : k Ψ = e i ω k t [ Ae ikx + Be ikx ] e iV 0 t / h = [ Ae ikx −ω k t ) + Be k t ) ] e iV 0 t/ h ( ( argument 2. How does point of constant phase move? const = kx φ k t x φ (t) =+ ω k t + x φ (0) moves in +x k direction if k > 0 revised 9/4/02 10:35AM
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*d 5.73 Lecture #3 3 - 3 h k 2 h k v (half as fast as we dx φ ω k 2m phase velocity v φ = 2m = 2 p m = 2 naively expect) v φ = dt = k ? first term in Ψ (x,t) moves to +x (right), second to –x (left). But if we treat the e iV 0 t/ h = e i ω 0 t term explicitly, we get v φ = ω k + ω 0 ! Any velocity we want! IS THIS A PROBLEM? WHY NOT?
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notes03 - 5.73 Lecture #3 3-1 Reading Chapter 1, CTDL,...

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