hw3 - Homework Week 3 Due Friday October 16 Chemistry 110A...

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Homework Week 3: Due Friday, October 16 Chemistry 110A — 2009 Professor McCurdy 1. Problem 4-25 in McQuarrie and Simon – The algebra should be very short and it’s a good idea to do this one before problem 2. 2. For the particle-in-box, the system is prepared in the state described the time- dependent wave function " x , t ( ) : " x , t ( ) = N # 1 x ( ) e $ iE 1 t / h + # 2 x ( ) e $ iE 2 t / h ( ) where " n x ( ) = 2 L sin n # x L $ % & ( ) and E n = n 2 " 2 h 2 2 mL 2 are the normalized particle-in-a-box functions and their corresponding energies. (a) Find the normalization constant N, and show that it does not depend on time. (b) Evaluate the expectation value of the energy E . (c) Evaluate the expectation value of position x to show that it is x t = L 2 " 16 L 9 # 2 cos E 2 " E 1 h t $ % & ( ) 3. Problem 4-26 in McQuarrie and Simon. Hint: This one is a little tricky at one point. To get to that point you just differentiate the integral that defines x using the product rule. Remember that time is real when you do that, so " " t # * = " " t # $ % & ( ) * , and use the time-dependent Schrödinger equation. Also
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