- #1
snoopies622
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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.) 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.
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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