Postulate 4: Time Evolution

Measurement (Postulate 3) is one way a state can change. Between measurements, a closed quantum system evolves smoothly and deterministically. Postulate 4 specifies that evolution.

The time-evolution postulate

Postulate 4. The state of a closed quantum system evolves in time by a unitary operator. If the state is $|\psi(t_1)\rangle$ at time $t_1$ and $|\psi(t_2)\rangle$ at time $t_2$ , then there is a unitary $\hat{U}(t_2, t_1)$ such that
$|\psi(t_2)\rangle = \hat{U}(t_2, t_1)\,|\psi(t_1)\rangle, \qquad \hat{U}^{\dagger}\hat{U} = \hat{I}.$

A unitary operator preserves inner products: $\langle\hat U\phi|\hat U\psi\rangle = \langle\phi|\psi\rangle$ . In particular it preserves norms, so a normalized state stays normalized — probabilities continue to sum to one for all time. Unitary evolution is also reversible: since $\hat{U}^{-1} = \hat{U}^{\dagger}$ exists, you can always run the dynamics backward, in sharp contrast to the irreversible collapse of measurement.

The continuous form: the Schrödinger equation

The discrete statement above has a differential counterpart. For a continuous-time evolution generated by a Hermitian Hamiltonian $\hat{H}$ (the energy observable), the state obeys the time-dependent Schrödinger equation:

i\hbar\,\frac{d}{dt}\,|\psi(t)\rangle = \hat{H}\,|\psi(t)\rangle.

When $\hat{H}$ is time-independent, this integrates to

|\psi(t)\rangle = e^{-i\hat{H}t/\hbar}\,|\psi(0)\rangle, \qquad \hat{U}(t) = e^{-i\hat{H}t/\hbar}.

The operator $\hat{U}(t) = e^{-i\hat{H}t/\hbar}$ is unitary precisely because $\hat{H}$ is Hermitian: $\hat{U}^{\dagger} = e^{+i\hat{H}t/\hbar} = \hat{U}^{-1}$ . This is the deep reason observables are Hermitian and dynamics are unitary — the two are linked by exponentiation.

Energy eigenstates evolve by a phase

If $|E_n\rangle$ is an eigenstate of the Hamiltonian, $\hat{H}|E_n\rangle = E_n|E_n\rangle$ , then it evolves by a pure phase:

e^{-i\hat{H}t/\hbar}|E_n\rangle = e^{-iE_nt/\hbar}|E_n\rangle.

Such stationary states have time-independent measurement statistics (the global phase is unobservable). A general state is a superposition $\sum_n c_n|E_n\rangle$ , and each component picks up its own phase $e^{-iE_nt/\hbar}$ . The relative phases between components do change, and that is what produces nontrivial dynamics and interference.

Gates are unitaries

In the circuit model every gate is a unitary operator — a particular choice of $\hat{U}$ . The single-qubit rotation

R_z(\theta) = e^{-i\theta \hat{Z}/2} = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}

is literally $e^{-i\hat{H}t/\hbar}$ for the Hamiltonian $\hat{H} = \tfrac{\hbar\theta}{2t}\hat{Z}$ . The Hadamard, Pauli, and controlled gates of the simulator are all unitaries; running a circuit is applying Postulate 4 step by step. Because each gate preserves the norm, the state vector stays on the unit sphere throughout a computation.

Summary

Between measurements the world is unitary: deterministic, reversible, norm-preserving, and generated by the Hermitian Hamiltonian through the Schrödinger equation. Measurement is the only place where the smooth unitary story is interrupted by probabilistic collapse — the boundary between Postulate 4 and Postulate 3.