Homework Help: Compatibility theorem

1. Mar 14, 2017

davon806

1. The problem statement, all variables and given/known data
Please see the following,I am confused by the word "only".
2. Relevant equations

3. The attempt at a solution
I understand that the Compatibility theorem ensures we can find a basis of common eigenfunctions of $\hat{A} ,\hat{B}$.If each pair of eigenvalues {A_i,B_j} identifies uniquely one vector of the basis,then the set {$\hat{A} ,\hat{B}$} forms a CSCO.
Here they are asserting $\tilde{u_1} , \tilde{u_2}$ are the only eigenstates of B in the plane.I don't see the reason for that.Could we find more orthonormal eigenstates of B in the plane spanned by the degenerate states?

2. Mar 14, 2017

Staff: Mentor

They are not degenerate with respect to B. That's exactly the case (1) discussed there, the two eigenvectors with respect to B are unique. If both eigenvalues of B are the same, see case (2).