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Compatibility theorem

  1. Mar 14, 2017 #1
    1. The problem statement, all variables and given/known data
    Please see the following,I am confused by the word "only". Q.jpg
    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 [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?
     
  2. jcsd
  3. Mar 14, 2017 #2

    mfb

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    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).
     
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