Position eigenstate ($\hat{q}|q⟩=q|q⟩$)

A generalized eigenstate of the position operator is labeled by its position eigenvalue. In a continuous position coordinate the normalization is distributional: $⟨q|q^{\prime }⟩=\delta (q-q^{\prime })$ and $\int dq|q⟩⟨q|=1$. Individual position eigenstates are not normalizable Hilbert-space vectors. The Dirac delta distribution normalization makes $⟨q_f|U|q_i⟩$ the integral kernel of a time-evolution operator, such as the free-particle propagator.