Compatibility Thm HW: Can We Find More Orthon Eigenstates?

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davon806
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Homework Statement


Please see the following,I am confused by the word "only".
Q.jpg

Homework Equations

The Attempt at a Solution


I understand that the Compatibility theorem ensures we can find a basis of common eigenfunctions of [itex]\hat{A} ,\hat{B}[/itex].If each pair of eigenvalues {A_i,B_j} identifies uniquely one vector of the basis,then the set {[itex]\hat{A} ,\hat{B}[/itex]} forms a CSCO.
Here they are asserting [itex]\tilde{u_1} , \tilde{u_2}[/itex] 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?
 
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davon806 said:
Could we find more orthonormal eigenstates of B in the plane spanned by the degenerate states?
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).
 
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