% miko:type: atomic
% miko:title: Klein-Gordon Equation
% miko:summary: The relativistic wave equation associated with the energy-momentum relation for spin-zero fields.
% miko:tags: particle
% miko:difficulty: undergrad
% miko:related: complete-particle-physics, dirac-equation, schrodinger-equation
% miko:keywords: relativistic wave equation, scalar field

\documentclass{article}
\usepackage{amsmath}
\title{Klein-Gordon Equation}
\begin{document}
\maketitle

\section{Derivation}

Start from

$$
E^2=p^2c^2+m^2c^4.
$$

Use the operator substitutions

$$
E\to i\hbar\frac{\partial}{\partial t},
\qquad
\mathbf p\to -i\hbar\nabla.
$$

Then

$$
\left(\frac{1}{c^2}\frac{\partial^2}{\partial t^2}
-\nabla^2
+\frac{m^2c^2}{\hbar^2}\right)\phi=0.
$$

\section{Interpretation}

In modern language this is the field equation for a free scalar field, not a
single-particle probability wave in the same sense as the Schrödinger equation.

\end{document}
