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The state of a quantum system is indicated by##\Psi## in Dirac notation.

Every observable (position, momentum, energy, angular momentum, spin, etc.) corresponds to a linear operator that acts on ##\Psi##.Every operator has its own set of eigenstates which form an orthonormal basis that can be used to expand the state ##\Psi##. The same state ##\Psi## can be represented also using the eigenstates of any other operator. Some operators have a continuous spectrum and some a discrete spectrum.

I understand that ##\Psi## lives in the vectors space of square integrable functions and that every linear vector space has an infinite number of possible bases all having the same number of basis vectors. Each bases contains as many linearly independent vectors as the dimension of the linear vector space. Among these bases, there is also the basis formed by the eigenstates of a specific operator.

Here my question: does the state ##\Psi##, which describes the system, live in many different vector spaces simultaneously and is each operator associated to each different vector space? For instance, the state ##\Psi## could be expanded as a weighted sum of several energy eigenstates which implies that the energy basis is multidimensional and the vector is in that case living in a multidimensional vector space. But if we consider, for instance, the spin operator, there are only two eigenstates so the basis is just a two dimensional basis and the state ##\Psi## becomes the weighted sum of those two eigenstates. The spin basis cannot live in the same vector space as the energy basis or the basis of any other operator...

How do things work?

Thanks!!

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# I State Representation in QM and Vector Spaces

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