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equations:klein-gordon_equation [2018/03/13 11:18]
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-====== Klein-Gordon Equation ======+<WRAP lag>​$ ​ ( \partial _{\mu} \partial ^{\mu}+m^2)\Phi ​0 $</​WRAP>​
  
 +====== Klein-Gordon Equation ​  ​======
  
-<note tip$$( \partial _{\mu} \partial ^{\mu}+m^2)\Phi = 0 $$ +<tabbox Intuitive
  
--->​Definitions#​+The Klein-Gordon equation describes how the state of a relativistic (= fast moving) quantum system without spin changes in time.
  
-  * $\partial _{\mu} $ denotes the partial derivative and $\partial _{\mu} \partial ^{\mu}$ stands ​for a sum using the Einstein sum convention, i.e. $\partial _{\mu} \partial ^{\mu} =  \partial _0 \partial^0 ​\partial _1 \partial^1 -\partial _2 \partial^2 -\partial _3 \partial^3$. +The analogous equation ​for systems with half-integer spin is the [[equations:​dirac_equation|]].
-  * $m$ denotes the mass of the particle. +
-  * $\Phi$ ​is either ​the wave function of the spin $0$ particle if we use the Klein-Gordon equation in a particle theory, or describes the spin $0$ field if we work in a field theory+
  
-<--+If the system only moves slowly, the Klein-Gordon equation becomes the [[equations:​schroedinger_equation]]. ​
  
-</​note>​ +  ​ 
-<​tabbox ​Why is it interesting?​+<​tabbox ​Concrete 
 +The Klein-Gordon equation can be derived from the Lagrangian
  
-The Klein-Gordon ​equation ​is the correct equation of motion that describes free [[basic_notions:​spin|spin]] $1$ particles.+\begin{equation} \mathscr{L}= \frac{1}{2}( \partial _{\mu} \Phi \partial ^{\mu} \Phi -m^2 \Phi^2) \end{equation}
  
 +using the [[equations:​euler_lagrange_equations|Euler-Lagrange equations]].
  
-<tabbox Layman> ​+----
  
-<note tip> +**Solutions** 
-Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during ​coffee break or at cocktail party+ 
-</​note>​ +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} 
-  ​ + 
-<tabbox Student> ​+----
  
   * A nice discussion can be found in [[http://​www.quantumfieldtheory.info/​website_Chap03.pdf |chapter 3 of Klauber'​s Student Friendly QFT book]]   * A nice discussion can be found in [[http://​www.quantumfieldtheory.info/​website_Chap03.pdf |chapter 3 of Klauber'​s Student Friendly QFT book]]
   * For an elementary derivation of the Klein-Gordon equation see Physics from Symmetry by Schwichtenberg   * For an elementary derivation of the Klein-Gordon equation see Physics from Symmetry by Schwichtenberg
 +
 +----
 +
 +**Graphical Summary**
 +
 +The diagram below shows the Klein-Gordon equation and its Lagrangian in various forms. For a more detailed explanation see [[https://​esackinger.wordpress.com/​|Fun with Symmetry]]. ​
 +
 +{{:​equations:​klein_gordon.jpg?​nolink}}
    
-<​tabbox ​Researcher+<​tabbox ​Abstract
  
 <note tip> <note tip>
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 </​note>​ </​note>​
  
---Common Question 1#+<tabbox Why is it interesting?​
  
-  +The Klein-Gordon equation is the correct equation of motion that describes free [[basic_notions:​spin|spin]] $0$ particles. For a spin-1 generalization see the Duffin-Kemmer-Petiau equation.
-<--+
  
---> Common Question 2# 
- 
-  
-<-- 
   ​   ​
-<​tabbox ​Examples+<​tabbox ​Definitions
  
---> Example1# 
  
-  +  * $\partial _{\mu} $ denotes the partial derivative and $\partial _{\mu} \partial ^{\mu}$ stands for a sum using the Einstein sum convention, i.e. $\partial _{\mu} \partial ^{\mu} =  \partial _0 \partial^0 - \partial _1 \partial^1 -\partial _2 \partial^2 -\partial _3 \partial^3$,​ 
-<--+  * $m$ denotes the mass of the particle, 
 +  * $\Phi$ describes the spin $0$ field if we work in a field theory.  
 +  * Note: $\Phi$ cannot be interpreted as a wavefunction because it is a real valued field; it is its own anti-particle like the Majorana fermion. Only in the case that it is the U(1)-charged (requires 2 independent real Klein Gordon fields that are symmetry transform into each other) is a naive wavefunction interpretation possible. Basically, you get a relativistic scalar superfluid field. Nevertheless,​ there are single particle wavefunctions lurking in the single real Klein-Gordon theory. But you need to use the coherent state representation to see the 1st quantized operators from the complex annihilation and creation operators. Essentially undoing the second quantization. ​
  
---> Example2:# 
- 
-  
-<-- 
-  ​ 
-<tabbox History> ​ 
  
 </​tabbox>​ </​tabbox>​
  
  
equations/klein-gordon_equation.1520936315.txt.gz · Last modified: 2018/03/13 10:18 (external edit)