Commutation The operators A and B commute: AB = BA [ A, B ] = 0. | Ψ a ± is an eigenstate of A with eigenvalue a . A | Ψ a ± = a | Ψ a ± Then AB | Ψ a ± = BA | Ψ a ± = aB | Ψ a ± A ( B | Ψ a ± ) = a ( B | Ψ a ± ) So B | Ψ a ± is an eigenstate of A with eigenvalue a . If the state is non-degenerate (unique eigenstate for a particular eigenvalue) then B | Ψ a ± must be proportional to | Ψ a ± so that B | Ψ a ± = b | Ψ a ± . Thus | Ψ a ± is a simultaneous eigentstate of A and B with eigenvalues a and b . We can call that state | Ψ a,b ± . If a system is in this state then it has deﬁnite values for the observables A and B . The deﬁnite values are a and b . The operators S x , S y , and S z do not commute; there is no state which is simultaneously an eigenstate of any two of these. The commutation realtions among the S operators is of the form [ S x , S
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