Here is the HOMEWORK for this week

1) Gas' hint is good! A and D are arbitrary, meaning YOU can pick them however you like. (0 or nonzero, real or complex, and any equations involving them must still be correct)

A matrix, M, is unitary if the inverse of M is equal to the Hermitian Conjugate of M. The Hermitian Conjugate of M is defined to be the transpose of the complex conjugate of M. (The complex conjugate of a matrix is just where you complex conjugate every single element. The transpose of a matrix is where you flip about the diagonal, i.e. the ij'th element is switched with the ji'th element. The elements on the diagonal, of course "switch" with themselves, and thus stay the same)

2) To solve this problem, you must basically solve for the reflection and transmission coefficients of a delta function barrier. However, you should set it up just like in problem 1, with waves coming in from both left and right, and solve the whole problem at once. You can use continuity of the wave function at x=b, but remember you cannot use continuity of the derivative. (There is something different going on instead, see e.g. Gas eqn. 5.13) (Make use of the results of problem one to help you prove the S matrix is unitary!)

3) Gas 5-4. Ignore Gas' hints!! Just solve the problem directly. Look at Gas eq. 5-63 (the second one, since this is supposed to be the odd solution), and think about what you know already. When you are given the binding energy, think about what this is. (Is it E? -E? V0? E+V0?... Draw a little picture...) I assumed that "range" means the same thing as Gas' parameter "a" for his finite well problems. Your answer for V0 should come out to be some reasonable number of MeV, it turns out to be quite a bit more than 2.2 MeV (i.e, although the well is quite deep, the particle is still just barely bound.) This is a model for a deuteron!

4) The integral you get is tricky. Write it as a dimensionless integral before trying to evaluate it! Then look it up. I got something like (what are x1, 2, and 3?) Check the raw behaviour - as E -> 0, does your lifetime get longer or shorter? (What should it do?)

5) This is a good and important problem, which you should put some time into thinking about. No calculations are required! You should be able to sketch wavefunctions by thinking about general features - symmetry, number of nodes, curvature. Remember that curviness is related to energy. The sharper/curvier your function, the higher the energy. (The more tightly trapped, the higher the energy, by Heisenberg, right?) And, you know about wavefunctions for SINGLE wells, which may help too...