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equations:klein-gordon_equation [2018/03/29 13:56]
leot221 [Concrete]
equations:klein-gordon_equation [2021/03/31 18:22] (current)
edi [Concrete]
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 <tabbox Intuitive> ​ <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 formulasbut 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 slowlythe Klein-Gordon equation becomes the [[equations:​schroedinger_equation]].  
   ​   ​
 <tabbox Concrete> ​ <tabbox Concrete> ​
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   * 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 Abstract> ​ <tabbox Abstract> ​
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 <tabbox Why is it interesting?> ​ <tabbox Why is it interesting?> ​
  
-The Klein-Gordon equation is the correct equation of motion that describes free [[basic_notions:​spin|spin]] $1$ particles.+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.
  
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   * $\partial _{\mu} $ denotes the partial derivative and $\partial _{\mu} \partial ^{\mu}$ stands for a sum using the Einstein sum convention, i.e. $\partial _{\mu} \partial ^{\mu} =  \partial _0 \partial^0 - \partial _1 \partial^1 -\partial _2 \partial^2 -\partial _3 \partial^3$,​   * $\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,   * $m$ 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. ​+  * $\Phi$ ​describes ​the spin $0$ field if we work in a field theory.  
 +  * Note: $\Phicannot 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 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
  
  
equations/klein-gordon_equation.1522324578.txt.gz · Last modified: 2018/03/29 11:56 (external edit)