= Unitary time evolution
{title2=$U(t)=e^{-iHt/\hbar}$}
A time-independent Hamiltonian evolves a state by the unitary operator $U(t)=e^{-iHt/\hbar}$.
For a closed <quantum system>, <unitary time evolution> changes a <quantum state> through a <unitary operator>, preserving <norms> and <inner products>. For a time-independent <self-adjoint> <Hamiltonian operator> $H$, choose the initial time as zero:
$$
i\hbar\frac{d}{dt}|\psi(t)\rangle=H|\psi(t)\rangle,\qquad
|\psi(t)\rangle=U(t)|\psi(0)\rangle,\qquad U(t)=\exp(-iHt/\hbar).
$$
Here $\hbar=h/(2\pi)$ is the reduced <Planck constant>. The same $U(t)$ acts on every initial <quantum state>; it depends on the <Hamiltonian operator> and elapsed time, rather than the chosen preparation.<Quantum Physics II (8.05), Fall 2013/Quantum Dynamics>[Quantum Physics II (8.05), Fall 2013; Zwiebach, Quantum Dynamics, sections 2.1 and 2.3, printed pp. 10–11 and 15–16, equations (2.3), (2.6)–(2.7) and (2.45)]{ref}
The <self-adjointness> of $H$ makes the exponential a <unitary operator>. In finite <dimension>, conjugating its <matrix exponential> gives
$$
U(t)^\dagger=\exp(iHt/\hbar)=U(-t),\qquad
U(t)^\dagger U(t)=UU^\dagger=I.
$$
The two exponents commute because both are multiples of $H$. Hence, for any two initial <vectors>,
$$
\langle U\phi|U\psi\rangle=\langle\phi|U^\dagger U|\psi\rangle
=\langle\phi|\psi\rangle,\qquad \|U\psi\|=\|\psi\|.
$$
This short proof explains why normalization survives, why <orthogonality> of initial <vectors> survives, and why the <transition probability> between two states evolved by the same $U$ is unchanged. It does not require that a fixed <quantum measurement> have time-independent <probabilities>: its measurement <basis> need not move with the <quantum states>.
For an unbounded <self-adjoint operator>, define the exponential through the <spectral theorem>, rather than assuming that its power series acts on every <vector>. It is a <unitary operator> on the entire <Hilbert space>; the strong <Schrödinger equation> holds for initial <vectors> in the <operator domain> $D(H)$. For a <density operator>, <unitary time evolution> reads $\rho(t)=U(t)\rho(0)U(t)^\dagger$. It preserves <eigenvalues> and <purity of a density operator>, but can change its entries in a fixed <basis>.
The useful distinction is between conserving total <probability> and conserving each measurement's <probabilities>. A fixed <energy> spectrum supplies different ticking rates for different <probability amplitudes>. Their relative <quantum phases> can move the <Bloch vector> and create <quantum interference> even while every <energy eigenstate> population stays fixed. <Qubit precession under a constant Hamiltonian> makes this visible; <Ramsey interferometry> converts the accumulated relative <quantum phase> into a population difference that a simple readout can detect.
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