Quantum Mechanics Operators, Hermitian and Eigenvalues

[TtotheBo]

1.

a) The action of the parity operator, $$\Pi$$(hat), is defined as follows:

$$\Pi$$(hat) f(x) = f(-x)

i) Show that the set of all even functions, {e_n(x)}, are degenerate eigenfunctions of the parity operator. What is their degenerate eigenvalue? The same is true for the set of all odd functions, {O_n(x)}, but their degenerate eigenvalue is different from that of the even eigenfunctions, what is it?

ii) Show that the parity operator is a hermitian operator

iii) We know that eigenfunctions of a hermitian operator whose eigenvalues are different are orthogonal. What important integration result does this imply for the functions you consider in part i) above?

b) Suppose f(x) and g(x) are two degenerate eigenfunctions of an operator Q(hat) that have the same eigenvalue, q. Show that any linear combination of f(x) and g(x) is itself an eigenfunction of Q(hat) with eigenvalue, q.

Homework Equations

Nothing out of the ordinary.
See attempt at a solution.

The Attempt at a Solution

First of all, I suck at QM

I know Hermitian Operators have real eigenvalues

I know Eigenfunctions corresponding to different eigenvalues are orthogonal
Integral of q_n(x)q_m(x) dx = 0

I think know that to prove it's Hermitian in part ii), I need to get the LHS equal to the RHS beginning with:

The Integral of $$\Phi$$* ($$\Pi$$(hat)) $$\Psi$$ dx
=
(The Integral of $$\Psi$$* ($$\Pi$$(hat)) $$\Phi$$ dx)*I know that degenerate means E_n = E_m
No clue where to even begin though.

I'm not sure at all where to begin with this question.. Perhaps somebody could give me a push in the right direction? Even the smallest hints as to where to begin would be fantastic. Then I can try myself and come back later with more questions if needs be.
Thank you

 

[Mindscrape]

i) In this case, don't think of degenerate "energies," think about degenerate eigenvalues. Energy is an eigenvalue of the hamiltonian operator, not the parity operator.

ii) Your definition of hermitian would be wrong is you did this. Review hermitian operators again.

iii) I don't know how to hint this without giving it away. The question basically already gives away the answer.

b) Give it a try.