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?
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?
