Normalized state vector

A normalized state vector in a complex Hilbert space $H$ is a vector $\psi\in H$ with $\lVert\psi\rVert=1$. It lies on the unit sphere.

In quantum mechanics, $\psi$ and $e^{i\theta}\psi$ represent the same pure state, because all transition probabilities depend only on squared inner-product magnitudes. The phase-independent state is equivalently represented by the rank-one projector $|\psi\rangle\langle\psi|$.