Pure state versus superposition

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I am a little bit versed in the formalism of Hilbert space, the state vector and the matricies and their eigenvector basis of observables in QM. I understand that a pure state in one observable basis may be (is) a superposition in another basis (spin xUP) is superposition of spinyUP, spin yDOWN eigenvectors. And that a superposition may also be a pure state if you pick the right observable. My question: is EVERY allowable superposition in quantum mechanics, every state the system could end up in, a pure state for some observable, even if this observable is very complex to realize? I'm speaking in principle rather than practice. If so, would it be correct to say that the state of the system is always in principle an observable pure state but we just have to keep changing bases? I apologize if I'm wording things awkwardly.
 

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Hurkyl
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In the formulation you're familiar with, every normalized vector represents a pure state. (Of course, if two such vector are a multiple of each other, they represent the same state)

Conversely, every pure state can be represented by a normalized vector.

Also, every vector (or nonzero vector, depending on the details of how you define things) is indeed an eigenvector for some observable.



There are other states, though. These are not formed by superimposing, but by mixing -- if you represent states as density matrices, a linear combination of states (with positive real coefficients that add to 1) usually gives a mixed state.



Maybe I learned things oddly, but I've never heard of a "pure state for an observable".
 
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Clear. I'm thinking about a state that evolves in time, for example. Like if you had the square well problem but your system was in more than one energy state, so the state evolves as the mean value sloshes back and forth in time. Is that a superposition of say pure energy eigenstates or is it a mixed state? Are mixed states also pure states for some outlandish observable?
 
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Wait, I think I see from some other postings. The mixed states really are a different case, more like statistical mechanics.
 
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Hurkyl
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Clear. I'm thinking about a state that evolves in time, for example.
Vectors in the Hilbert space (and in the domain of the Hamiltonian) evolve to vectors in the Hilbert space, right? So....
 
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A. Neumaier
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is EVERY allowable superposition in quantum mechanics, every state the system could end up in, a pure state for some observable, even if this observable is very complex to realize?
Yes. If \psi is a pure state then it is an eigenstate of the projector P = \psi\psi^* with eigenvalue 1, and P is an observable according to the standard definition (self-adjoint). Moreover, if the Hilbert space is low-dimensional, P can be realized quite well.
 

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