Angular momentum commutators

• I
Hi.
To show that [ L2 , L+ ] uses the following commutators [ L2 , Lx ] = 0 and [ L2 , Ly ] = 0 . But if [ L2 , Lx ] = 0 this shows that L2 and Lx have simultaneous eigenstates ; but then should L2 and Ly not commute ?
Thanks

Nugatory
Mentor
Hi.
To show that [ L2 , L+ ] uses the following commutators [ L2 , Lx ] = 0 and [ L2 , Ly ] = 0 . But if [ L2 , Lx ] = 0 this shows that L2 and Lx have simultaneous eigenstates ; but then should L2 and Ly not commute ?
Thanks
No. If A and B commute so have a common eigenbasis and B and C commute so have a different common eigenbasis, it does not follow that A and C commute and have a common eigenbasis.

dyn
Thanks. So L2 and Lx have a common eigenbasis as the 2 operators commute and L2 and Ly have a different common eigenbasis. But as Lx and Ly do not commute these 2 sets of eigenbases can never be the same ?

PeroK