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I'm sure trivial and I can figure out by looking closely, but what is the expectation value of the partial derivative, with respect to time, of momentum operator?

This is the problem that was in the homework and the exam. <math>\frac{\partial <p>}{\partial t} = \left< - \frac{\partial V}{\partial x} \right></math>. The way you worded your question, however, seems to be asking about <math>\left< \frac{\partial p}{\partial t} \right></math>, which I assume would be the same, but I don't know.

Yes, the second way is what I have in mind. I think it is different though, because the partial is already inside the integrand and does not operate on conjugate wave function. I will look, but I think it is zero?

I realize I forked my own thread the other day, and I am still quite curious about this - what intuitive methods, general techniques, and approximations could be used to finish the first exam in one hour or less? My re-write took 13 pages and 6 hours or something like that. Yet somebody got a 98 the first time through. Perhaps that person would like to share their thoughts? Broadly, I think this is a very important question for the class. The mean was about 42. I'm happy to say I got a 43. And I was writing pretty much the whole time. If I recall, no one left early. I invite anyone who got a 98, or close to it, to help the rest of us out. Thanks and please, East End

I agree, the test was much too long for many people to finish. I also wrote as fast as I could the entire time. What gave me trouble was the fact that we had a “think outside the box” type of problem - problem 3. Although it closely paralleled the homework/book examples, the fact that it was slightly different required a little “thinking” time, which I had no time for, since I didn't have enough time to write the easier problems too which I definitely knew the answers. When correcting the exam on my own time, I found the problem not difficult, but just required extra thought. Also, a suggestion: allow us to have trig identities. Even if we memorize them, we may not have time to check to see we didn't make a mistake. Suggestion to East End: reviewing the book, homeworks, and discussion problems helped me. Just make sure you know the basic concepts of each significant topic we cover.

You were definitely missing some shortcuts if you took 13 pages. In problem 1, the key was the trigonometric identity <math>sin^2(x)=\frac{1-cos(2x)}{2}</math>. Then all you have to do is integrate the RHS of that equation for <math><p^2></math>. <math><x^2></math> is a little trickier. Multiply RHS of the identity by <math>x^2</math>. The first term is trivial. The second term is a classic application of integration by parts, which needs to be done twice. A lot of the terms in the sum you end up with will vanish due to the limits of integration - they will be products of x and a sine or cosine of 2x. The lower limit will be 0, causing any terms containing x to vanish, and the trig functions will be 1 or 0, at one or both limits.

To make it easier to read (and write) I dropped the constant coefficients.

This would be a fantastic homework problem. It's actually in the book (2.11). It would be a good test problem too, if we were allowed to spend more time on it, AND we were given that trigonometric identity. There are countless tricks in calculus, trigonometry, and algebra that we have all learned, and we can't possibly have all of them on tap at all times. I am not a physics student, so perhaps my experience is not on par with the class, but I certainly have not had to integrate enough squares of trig functions for a double-angle formula to pop into my head as soon as I see it. Maybe it's a very common thing in QM? Us students don't know that yet. If we did SEVERAL problems using this trick for homework/discussion, we would wise up and memorize it. For example, problem 4 was pretty tricky, but we did it as homework, as well as covering it in discussion (for my section at least), so I think it's perfectly reasonable to expect us to know the tricks for that (again, it's a somewhat long derivation - time was still tight). Having notecards will not alleviate this problem, as we can't write down the tricks if we don't know they're important.

I think the best improvement that can be made on the quiz is to specifically give us the tricks we need to use to solve each problem, if they have not been thoroughly covered in class. Seeing a double-angle formula printed on a piece of paper is not going to hurt our intuitive understanding of quantum mechanics.

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