a) This is very easy if you exploit the orthonormality of !

(No actual integrations need to be done)

b) You can just write the answer down, no calcluations needed here either!

c) This part is a pain, and involves at least one hard integral. (There are several integrals to do, but some of them you may be able to use symmetry to guess the answer!)

e) There's an easy quick way to do this one, and a much harder way. Choose the easy way!! (You might want to express all answers as some number times E_1)

f) You can write your frequencies symbolically, in terms of E_1, if you want.

2) Gas 4-6a) You do NOT have to do a bunch of integrals yet! A thoughtful answer in (mostly) words is perfectly sufficient here.

For b) See the hints from the last problem set, for Gas. 4-5. This time, the well goes from x=-a/2 to a/2. (Again, you should be able to guess the mathematical form of the wave functions, but be sure you get your normalization right)

For this part, you will have to do a couple of integrals, but they should be really easy ones.

3) (Gas 4-9) 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)

4) This problem is very similar to our first 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!