Basis for state space


Remedial Math Student
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According to conventional quantum mechanics, given a quantum system, everything that can be known about it - its state - is given by a vector, a state vector. In standard linear algebra, any vector is an element of a vector space. And again, theory states that for any such space there exists a subspace called a basis for the space, such that every vector in the space can be written as a linear combination of the basis vectors.

For definiteness, I write, for the arbitrary vector $$\psi$$ that $$\psi = \sum_{jk}\alpha_j\varphi^j\alpha_k\varphi^k$$ (the $$\varphi$$ are the basis vectors, the $$\alpha$$ are scalars).

Now looking at the sum above, this looks awfully like a mathematical representation of so-called superposition of states. So my question is this: if this is correct, does that imply that most quantum systems are in superposition of what might be called "singular states" ?

And if it's not correct, what is the basis for state space?
Yes, most quantum systems are typically in a superposition of these “singular states” or eigenstates.

As for the basis of the state space, it’s typically chosen to be the set of eigenstates of an observable that’s relevant to the system.