Klein-Gordon Equation [The Physics Travel Guide]

--- equations:klein-gordon_equation [2018/03/29 13:55]
leot221 [Concrete]
+++ equations:klein-gordon_equation [2018/04/16 09:12]
jakobadmin [Intuitive]
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 <tabbox Intuitive>
-<note tip>
+The Klein-Gordon equation describes how the state of a relativistic (= fast moving) quantum system without spin changes in time.
-Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
-</note>
+The analogous equation for systems with half-integer spin is the [[equations:dirac_equation|]].
+If the system only moves slowly, the Klein-Gordon equation becomes the [[equations:schroedinger_equation]].
 <tabbox Concrete>
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 using the [[equations:euler_lagrange_equations|Euler-Lagrange equations]].
+----
+**Solutions**
+The most general solution of the Klein-Gordon equation is\begin{equation}\label{KGsol} \Phi(x)= \int \mathrm{d }k^3 \frac{1}{(2\pi)^3 2\omega_k} \left( a(k){\mathrm{e }}^{ -i(k x)} + a^\dagger(k) {\mathrm{e }}^{ i(kx)}\right) .\end{equation}
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