% miko:type: atomic
% miko:title: Quantum Harmonic Oscillator
% miko:summary: The exactly solvable quantum system underlying vibrations and field modes.
% miko:tags: quantum
% miko:difficulty: undergrad
% miko:related: schrodinger-equation, commutator, small-oscillations
% miko:keywords: ladder operator, oscillator, zero-point energy

\documentclass{article}
\usepackage{amsmath}
\title{Quantum Harmonic Oscillator}
\begin{document}
\maketitle

\section{Hamiltonian}

$$
H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2.
$$

\section{Energy Spectrum}

The allowed energies are

$$
E_n=\hbar\omega\left(n+\frac{1}{2}\right),
\qquad n=0,1,2,\dots.
$$

The ground state has nonzero zero-point energy.

\end{document}
