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I am confused with the concept of state of a quantum system. First postulate of QM ussualy says that the wave function of the system contains all information about the state of the system. But reading Dirac's Principles of QM made me think that state of the system always corresponds to some specific observable we want to measure. For example consider spin-1/2 particle. If We are interested in the spin in z-direction then corresponding state $| \psi \rangle$ of the system is a superposition of all eigenvalues of the operator $S_z$, i.e. $| \psi \rangle = c_1 | \uparrow \rangle + c_2 | \downarrow \rangle$ for some $c_1,c_2 \in \mathbb{C}$. So corresponding state space of such particle is two-dimensional Hilbert space $H = $Span$\{ | \uparrow \rangle , | \downarrow \rangle \}$. And if we were interested in some different observable, we might have different state space. So it appears to me that state of a quantum system always corresponds to some specific observable - is that correct?

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    $\begingroup$ Hi AlanHarper, and welcome to Physics Stack Exchange! We prefer to have one question asked per post, so could you perhaps remove the second question from this post and put it up separately? $\endgroup$ – David Z Mar 13 '14 at 16:47
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Spin is one observable, and it's space is spanned by two vectors, as you point out. What about all the other possible observables? Each observable has it's own "private" (as it were) Hilbert space. The state of the system as a whole is the direct product space of all those individual Hilbert spaces. The total state vector contains all information about all possible observables. The story of what a direct product space is, and how measurements of a individual observable are considered in these spaces is a long-ish story, so I'll wait to see if you have questions about that.

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  • $\begingroup$ Thank you for your answer. That is exactly what I am concerned about - what about all those observables! Once I realized that different observables may have different number of eigenstates (thus these eigenstates have to span different state spaces), my confusion about measurement and expansion of the state in basis eigenstates of the corresponding observable started. Could you provide more information about the direct product space, please? $\endgroup$ – AlanHarper Mar 13 '14 at 17:25
  • $\begingroup$ I'm going to renege slightly on my offer. It takes a quite a bit of notation, which I can't offer right now. But the idea is not that complicated. I will offer a link. It was surprisingly difficult to find a simple presentation that would answer your question. this link is a start, but it doesn't show how calculation of, e.g., expectation values works out. If I find a good link, I'll post it. $\endgroup$ – garyp Mar 13 '14 at 19:17

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