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This pertains to the quantum mechanics of angular momentum, so I'll ask it here:

If operators A and B commute, does it follow that

(i.)

or that

(ii.) merely some of them are?

According to Gillespie (see below) the commutivity of A and B implies that they "possess a common eigenbasis" (and the converse) and his proof seems to imply (i.), but it assumes that the operators are non-degenerate, and I don't know if the angular momentum operators are. Also, if (i.) and its converse are true, then it would seem to me that if A and B commute and A and C commute, then B and C would also commute, and I know that that's not supposed to be true. Nevertheless, so far I haven't found anyone explicitly say that (ii.) is the case.

The book is "A Quantum Mechanics Primer" by Daniel T. Gillespie, 1970.

If operators A and B commute, does it follow that

(i.)

*all*of A's eigenvectors are eigenvectors of B and vice versa,or that

(ii.) merely some of them are?

According to Gillespie (see below) the commutivity of A and B implies that they "possess a common eigenbasis" (and the converse) and his proof seems to imply (i.), but it assumes that the operators are non-degenerate, and I don't know if the angular momentum operators are. Also, if (i.) and its converse are true, then it would seem to me that if A and B commute and A and C commute, then B and C would also commute, and I know that that's not supposed to be true. Nevertheless, so far I haven't found anyone explicitly say that (ii.) is the case.

The book is "A Quantum Mechanics Primer" by Daniel T. Gillespie, 1970.

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