state vector

A state vector is the most abstract way to capture a quantum state: not as a function of position, but as a single arrow in an abstract space of possibilities, usually written |ψ⟩ in Dirac's bracket notation. Just as an ordinary vector can be described by its components along chosen axes, a state vector can be described by its amplitudes along a chosen set of basis states — and the wavefunction ψ(x) is simply one such list of components, the ones along the position axes.

Casting the state as a vector is powerful because it frees the physics from any one description. The same |ψ⟩ can be written out in positions, in momenta, in energy levels — different 'axes' for the same arrow — and switching between them is just a change of basis. Adding state vectors gives superposition; the angle between two of them, measured by their inner product, encodes how much they overlap and hence transition probabilities between them.

The space these arrows live in is a Hilbert space, and the rules are stricter than for everyday vectors: the amplitudes are complex, and physical states are taken to have length one, reflecting that total probability is one. An overall phase or length carries no physical meaning, so it is really the direction of |ψ⟩ that matters. This geometric picture, due largely to Dirac, unifies wavefunctions, spin states, and every other quantum system under one clean language.