Here is the HOMEWORK for this week

Almost all of the "proofs" and "show thats" this week are not meant to be hard! Most (except 6-4, see below) are only one (or two) liners!

Remember some basic features of Hermiticity:

For ANY operator O, the Hermitian adjoint is defined by (for all vectors psi and phi).

A consequence of the above is that if c is the operator "multiply by c",

(c is just some complex constant) then . (Convince yourself that this follows from the above!!)

If O is Hermitian, then .

For ANY operator O, it's true that , and ,

and . (Can you PROVE this last one? That would make a lovely exam question!)

If O is represented by a matrix, then

(In words, the Hermitian adjoint is the conjugate of the transpose)

Special hint for 6-4) Gas' hint is tricky: is complex, so are independent! I have a much simpler hint that should work better for you. Suppose you define the new quantity . It must be true that (because the norm of any vector is always positive definite). Work out the consequences of this fact, and you should be able to show the Schwarz inequality immediately.

#5) "Find the eigenvectors and eigenvalues of O" means solve the equation , where u is a vector, i.e. .

(You should find two non-trivial eigenvalues in this problem)

Normalizing a vector means , or writing out components in gory detail: