On studying quantum states of the hydrogen atom, I came with a question: what is the precise definition of a quantum state? Particularly, when solving the Schroedinger eq for the H using separation of variables, the quantum numbers and wavefunctions for the states are clear. However, mathematically speaking, any combination (linear) of wavefuntions is still a solution for the H atom; is this solution a state? Moreover, what are the quantum numbeers of this solution? It blows my mind, since this solution has 6 quantum numbers, instead of 3.
More generally, my question is: is there a precise definition of state of system?