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Uniqueness of eigenvector representation in a complete set of compatible observables
Sakurai states that if we have a complete, maximal set of compatible observables, say A,B,C... Then, an eigenvector represented by |a,b,c....> , where a,b,c... are respective eigenvalues, is unique. Why is it so? Why can't there be two eigenvectors with same eigenvalues for each observable? Does maximality of the set has some role to play in it?
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Answer
Yes, since it is the maximal set of compatible observables, it includes all observables for which $|a\rangle$, $|b\rangle$, $|c\rangle$, etc. are the eigenvectors (I'll use the notation $|\psi_1\rangle$, $|\psi_2\rangle$, $|\psi_3\rangle$ etc instead). Hence this includes the observable $D = \sum_k k |\psi_k\rangle \langle \psi_k|$ . However $D$ has a unique set of eigenvectors, and hence so does the any compatible set of observables which contains $D$.
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