Finding normalized eigenfunctions of a linear operator in Matrix QM

[Dixanadu]

Homework Statement

Hey everyone!
The question is this:
Consider a two-state system with normalized energy eigenstates \psi_{1}(x) and \psi_{2}(x), and corresponding energy eigenvalues E_{1} and E_{2} = E_{1}+\Delta E; \Delta E>0

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

Homework Equations

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

The Attempt at a Solution

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

 

[dextercioby]

I think the hint I can give is to use that S is linear and consider a linear combination of the 2 vectors psi1 and psi2.

 

[Dixanadu]

But is my matrix correct for S..?

 

[dextercioby]

Who cares, you needn't use any matrix to solve the problem.

 

[Dixanadu]

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.