Q: Function of an observable (QM) I'm reading Dirac's book about QM: I reached the point where he talks about function of observables (page 41 on my edition). He draws the conclusion that it is convenient to define a function of an observable $\xi$ as:
$$ f(\xi)|\xi' \rangle = f(\xi')|\xi' \rangle $$
He says that this expression is self consistent when applied to eigenkets $|\xi'\rangle$ which are not independent. And here comes the first question: why he says this? He continues: "if we have an eigenket $| \xi' A \rangle$ dependent on other eigenkets of $\xi$, these other eigenkets must all belong to the same eigenvalue $\xi'$, otherwise we should have an equation of the type of (31), which we have seen it is impossible". The expression (31) which he is talking about is:
$$ 0 = \int | \xi' a \rangle d\xi' + \sum_s | \xi^s b \rangle $$
which he proved to be inconsistent. Here it comes the second question: how the fact that eigenkets on which depends $| \xi' A \rangle$ don't have not the same eigenvalue could result in the above expression?
