Physics 3220, Fall '96. HINTS for Homework 4 (Due Wed, Sept 25)

Here is the HOMEWORK for this week

1) The procedure to calculate [A,B] (where A and B are operators) is to first let the commutator operator on an arbitary function f, giving A(Bf)-B(Af), and then try to express this result in the form Cf, where C is some new operator. The answer is then C. (You don't pick any specific f(x), you must leave it arbitrary.)

(Note: what do you suppose parity does to the second derivative operator?)

2b) Watch your limits.

c) Think about the sign of your answer. Does it make sense?

d) This useful function is called the theta-step function...

3) (Gas 4-4) Are you a master of integration by parts yet? If not, you can of course look it up, or let MMA do it for you. (Note: I think Gas. meant for you to look at Eq. 4-26, not 4-28.)

4) Gas' formulas for square well wavefunctions always assume a well that goes from x=0 to x=a. But in this problem, the well goes from x=-a to x=a! So, your wavefunctions aren't quite the same as his. They're of course

very similar - so similar, that if you just sketch the wave functions or , you should be able to write down the formulae by inspection. You may want to check that you got your normalization right, though.

5) 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 comments for the previous problem. This time, the well goes from x=-a/2 to a/2. (Again, 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.

6) Use Gas' hint, it's essential! Remember,

(1-x)^5 = 1 - 5x + 10x^2 - 10x^3 + 5x^4 - x^5.

(You knew that, right?)