Almost all of the "proofs" and "show thats" this week are not meant to be hard! Many 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)

#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: