Homework Help: Finding normalized eigenfunctions of a linear operator in Matrix QM

1. The problem statement, all variables and given/known data
Hey everyone!
The question is this:
Consider a two-state system with normalized energy eigenstates [itex]\psi_{1}(x)[/itex] and [itex]\psi_{2}(x)[/itex], and corresponding energy eigenvalues [itex]E_{1}[/itex] and [itex]E_{2} = E_{1}+\Delta E; \Delta E>0[/itex]

(a) There is another linear operator [itex]\hat{S}[/itex] that acts by swapping the two energy eigenstates: [itex]\hat{S}\psi_{1}=\psi_{2}[/itex] and [itex]\hat{S}\psi_{2}=\psi_{1}[/itex]. Show that the corresponding normalized eigenfunctions of [itex]\hat{S}[/itex] are [itex]\phi_{1}(x)=\frac{1}{\sqrt{2}}(\psi_{1}(x)+\psi_{2}(x))[/itex] and [itex]\phi_{2}(x)=\frac{1}{\sqrt{2}}(\psi_{1}(x)-\psi_{2}(x))[/itex], with eigenvalues [itex]\lambda_{1}=+1[/itex] and [itex]\lambda_{2}=-1[/itex]

2. Relevant equations
- The regular eigenvalue/eigenvector stuff.
- In matrix mechanics, the wavefunction [itex]\psi_{n}(x)=C_{1}(1,0...n)^{T}+C_{2}(0,1,0...n)^{T}+...C_{n}(0....n)^{T}[/itex]

3. The attempt at a solution
So I've found the matrix associated with the linear operator [itex]\hat{S}[/itex] in the natural basis [itex]U_{1}=(1,0)^{T}, U_{2}=(0,1)^{T}[/itex]; which is:
http://imageshack.com/a/img266/5713/bfo6.jpg [Broken]
This gives the correct eigenvalues of +1 and -1. But I dont know how to find the eigenfunctions that the question is talking about. I'm not even sure if the matrix is right...

 

But is my matrix correct for S..?

 

what other ways are there that I can solve this? is it the bra-kets? And yea that matrix is wrong, it should be 0 along the diagonal and 1 otherwise. That's the only way it can be linear always and hermitian, which is the requirement.