Talk:PlanetPhysics/Klein Gordon Equation

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: Klein-Gordon equation %%% Primary Category Code: 03.65.Pm %%% Filename: KleinGordonEquation.tex %%% Version: 2 %%% Owner: invisiblerhino %%% Author(s): invisiblerhino %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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The Klein-Gordon equation is an equation of \htmladdnormallink{mathematical physics}{http://planetphysics.us/encyclopedia/PhysicalMathematics2.html} that describes spin-0 \htmladdnormallink{particles}{http://planetphysics.us/encyclopedia/Particle.html}. It is given by: \[ (\Box + \left(\frac{m}{\hbar c}\right)^2) \psi = 0 \] Here the $\Box$ symbol refers to the \htmladdnormallink{wave operator}{http://planetphysics.us/encyclopedia/DAlembertOperator.html}, or \htmladdnormallink{D'Alembertian}{http://planetphysics.us/encyclopedia/DAlembertian.html}, and $\psi$ is the wavefunction of a particle. It is a Lorentz invariant expression. \subsection{Derivation} Like the \htmladdnormallink{Dirac equation}{http://planetphysics.us/encyclopedia/DiracEquation.html}, the Klein-Gordon equation is derived from the relativistic expression for total \htmladdnormallink{energy}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}: \[ E^2 = m^2c^4 + p^2c^2 \] Instead of taking the \htmladdnormallink{square}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} root (as Dirac did), we keep the equation in squared form and replace the \htmladdnormallink{momentum}{http://planetphysics.us/encyclopedia/Momentum.html} and energy with their \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} equivalents, $E = i \hbar \partial_t$, $p = -i \hbar \nabla$. This gives (in disembodied operator form) \[ -\hbar^2 \frac{\partial^2}{\partial t^2} = m^2 c^4 - \hbar^2 c^2 \nabla^2 \] Rearranging: \[ \hbar^2(c^2 \nabla^2 - \frac{\partial^2}{\partial t^2}) + m^2 c^4 = 0 \] Dividing both sides by $\hbar^2 c^2$: \[ (\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}) + \frac{m^2 c^2}{\hbar^2} = 0 \] Identifying the expression in brackets as the D'Alembertian and right-multiplying the whole expression by $\psi$, we obtain the Klein-Gordon equation: \[ (\Box + \left(\frac{m}{\hbar c}\right)^2) \psi = 0 \]

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