Quick noob question: commutative of eigenstates

[Ykjh98Hu]

let L be angular momentum operator.

[L^2 , Lz] = 0
[L^2 , Lx] = 0 (I haven't prove this, but appearantly it's correct according to lecturer)

does it imply that [Lx , Lz] = 0?

this is just one interesting thoughts that cross my mind because I recalled that if 2 matrix [A,B] =0, A and B will have same eigenvectors (ie same basis that diagonalise them). Does this apply to above case because:

if L^2 and Lz = 0, we can spell ALL eigenstates of them.
then Lx suppose to share ALL those eigenstates since it commutes with L^2 too.

And...it violate uncertainty principle! (impossible.) so someone point out my error please! thanks =)

 

[Petr Mugver]

It is true that if A and B commute, then they have a common complete set of eigenvectors. So obviously if A and C commute, then they also have a complete set of common eigenvectors, but nobody says these two sets must be the same! This of course means that some of the eigenvalues of A must be degenerate, because otherwise the eigenvectors are uniquely determined. This is exactly the case of L^2, whose eigenvalues j = 0, 1/2, 1, 3/2, 2, ... show (at least) a j(j+1)-fold degeneracy. In fact L_z and L_x don't commute, generally: their commutator is iL_y, which is most of the times different from zero.

 

[Ykjh98Hu]

It is true that if A and B commute, then they have a common complete set of eigenvectors. So obviously if A and C commute, then they also have a complete set of common eigenvectors, but nobody says these two sets must be the same! This of course means that some of the eigenvalues of A must be degenerate, because otherwise the eigenvectors are uniquely determined. This is exactly the case of L^2, whose eigenvalues j = 0, 1/2, 1, 3/2, 2, ... show (at least) a j(j+1)-fold degeneracy. In fact L_z and L_x don't commute, generally: their commutator is iL_y, which is most of the times different from zero.

ah thanks for the clarification! i notice [Lx,Lz] !=0 and that's why i posted this. i must've forgot about degenerate eigenvalues.