User Tools

Site Tools


equations:klein-gordon_equation

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision
Previous revision
Next revision Both sides next revision
equations:klein-gordon_equation [2017/12/04 08:01]
127.0.0.1 external edit
equations:klein-gordon_equation [2021/03/31 18:22]
edi [Concrete]
Line 1: Line 1:
-====== Klein-Gordon Equation ======+<WRAP lag>​$ ​ ( \partial _{\mu} \partial ^{\mu}+m^2)\Phi ​0 $</​WRAP>​
  
-<tabbox Why is it interesting?> ​+====== Klein-Gordon Equation ​  ​======
  
-<note tip>​It'​s the correct equation of motion that describes free [[basic_notions:​spin|spin]] $1$ particles. +<tabbox Intuitive
-</note>+
  
-<tabbox Layman> ​+The Klein-Gordon equation describes how the state of a relativistic (= fast moving) quantum system without spin changes in time. 
 + 
 +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]]. ​
  
-<note tip> 
-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>​ 
   ​   ​
-<​tabbox ​Student+<​tabbox ​Concrete 
 +The Klein-Gordon equation can be derived from the Lagrangian 
 + 
 +\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]]. 
 + 
 +---- 
 + 
 +**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} 
 + 
 +----
  
   * 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>
Line 23: Line 44:
 </​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.txt · Last modified: 2023/04/02 03:24 by edi