Unitary Evolution

Unitary evolution is the time-development of a quantum state under the action of a unitary operator generated by the Hamiltonian. In quantum mechanics, unitary evolution is the rigorous expression of determinism: although measurement outcomes are probabilistic, the evolution of the state vector between measurements is fully determined and reversible. The unitary operator U(t) = exp(−iHt/ℏ) preserves the inner product structure of Hilbert space, ensuring that probabilities sum to one at all times.

This preservation is not merely a mathematical convenience; it is the quantum analogue of Liouville's theorem in classical mechanics. Just as classical phase space volumes are conserved under Hamiltonian flow, quantum state norms are conserved under unitary evolution. The tension between this continuous, reversible unitary dynamics and the discontinuous, irreversible collapse postulate remains the central unresolved wound of quantum foundations — a structural mismatch between the algebra of evolution and the phenomenology of observation.