How are these eigenfunctions obvious (by inspection)?

[ihatelolcats]

[PLAIN]http://img251.imageshack.us/img251/1050/quantume.png

taken from http://quantummechanics.ucsd.edu/ph130a/130_notes/node338.html

I see how psi_211 and psi_21-1 are eigenfunctions, because they are just 0.
I don't see how they got the other two (+/-).

Thanks in advance

 

[G01]

Well, this is the way I would inspect the equations:

You have two non-zero off diagonal elements in your matrix. So, you know you are going to have some non zero eigenvalues.

The non zero values fall in the positions of the matrix that connect $\phi_{200}$ and $\phi_{210}$. Thus, we know that (from previously solving a lot of matrix eigenvalue problems and noticing patterns) the eigenfunctions will be linear combinations of $\phi_{200}$ and $\phi_{210}$.

The off diagonal elements are equal, so the coefficients of $\phi_{200}$ and $\phi_{210}$ will be equal. We need to have normalize eigenfunctions so, we get the linear combinations given above.