% miko:type: atomic
% miko:title: Time Evolution Operator
% miko:summary: The unitary operator that advances a quantum state in time.
% miko:tags: quantum
% miko:difficulty: undergrad
% miko:related: schrodinger-equation, wavefunction-normalization, commutator
% miko:keywords: unitary, Hamiltonian, quantum dynamics

\documentclass{article}
\usepackage{amsmath}
\title{Time Evolution Operator}
\begin{document}
\maketitle

\section{Time-Independent Hamiltonian}

If $H$ is time independent,

$$
U(t)=e^{-iHt/\hbar}.
$$

\section{State Evolution}

The state evolves as

$$
|\psi(t)\rangle = U(t)|\psi(0)\rangle.
$$

Unitarity preserves normalization.

\end{document}
