PS3_v2 - Physics 115A Problem Set 3 Due Tuesday October 13...

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Physics 115A, Problem Set 3 Due Tuesday, October 13 @ 5pm in the box outside Broida 1019 (PSR) Suggested reading: Griffiths 2.1-2.3 1 E and V for normalizable solutions Griffiths Problem 2.2: Show that E must exceed the minimum value of V ( x ), for every normalizable solution to the time-independent Schr¨ odinger equation. What is the classical analog to this statement? Hint: Rewrite Equation 2.5 in the form d 2 ψ dx 2 = 2 m ~ 2 [ V ( x ) - E ] ψ ; if E < V min , then ψ and its second derivative always have the same sign –argue that such a function cannot be normalized. 2 Initial States in a Box You prepare a particle in the infinite square well (with walls at x = 0 , a ) in an initial state described by a linear combination of two stationary states, Ψ( x, 0) = A [ ψ 1 ( x ) + ψ 3 ( x )] 1. Normalize Ψ( x, 0), i.e., find A . 2. Find Ψ( x, t ) and | Ψ( x, t ) | 2 . Express the latter as a sinusoidal function of time in terms of the variable ω = π 2 ~ / 2 ma 2 . 3. Compute h x i . Is this an interesting function of time? What happened to the time- dependent terms? Why? 4. Compute h p i . 5. If you measured the energy of the particle, what values might you get, and what is the probability of getting them? What is the expectation value of H
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