Here is the HOMEWORK for this week

1) I found that using Euler's theorem right away, in the form

gives me an integral that is just a sum of two pure exponentials, which are straightforward to integrate. Don't forget that you begin by calculating phi(p), but what you want in this problem is the absolute square, |phi(p)^2|. (And remember that |z|^2 = z z*) I found this problem to be the longest of this set.

I let MMA do some sketches for me, it really helped!

The answer you get (which Gas gives you) for phi^2 looks, superficially, like it has a zero in the denominator for some special value of p. But, at that value of p, the numerator is also vanishing, so in fact your function doesn't blow up there. You have to use L'Hopitals rule (twice, because of the square downstairs) to find out the limit. You might find it useful to do this when you're looking at what happens as n goes to infinity. Why? Because when n gets large, you might at first convince yourself that your function should be vanishing (like 1/n^2) But that can't be right, because you know that your function is normalized to one! So, there must be some place where it doesn't vanish. The logical place to look would be where the denominator is vanishing, naturally...

2) This is basically just a Fourier transform of a Gaussian problem, which you've now done repeatedly. (Gas Eq's 2-2 through 2-4 tell you how these integrals go, if you've forgotten) Last week's homework tells you what the width of Gaussian's are, if you've forgotten that....

3) This problem is very similar to our quiz, so look back at the solutions to the quiz if you're having trouble thinking about the interpretation part.

When calculating j, the answer may start out looking rather messy, but in the end the answer should look VERY simple, and rather similar to what you got on the quiz!

4) A Hermitian operator means the expectation value should be real.

(You might find it useful to change variables in your integrals, letting the new integration variable z = -x...) For the second part, think about symmetry (odd and even functions)! What's the integral of an odd function, from -infinity to +infinity?