I recently read that a quantum state is actually defined by a ray and not a vector. That is it is possible to multiply a state $\psi$ by any complex number $c\in \mathbb{C}$ and you won't be changing the physics in any way. I understand this mathematically, but I don't understand what the physical meaning of such an "equivalent state" would be since the new state need not be normalised if $c$ is not of the form $e^{i\phi}$.