equations:klein-gordon_equation

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equations:klein-gordon_equation [2018/03/13 11:15] jakobadmin |
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- | ====== Klein-Gordon Equation ====== | + | <WRAP lag>$ ( \partial _{\mu} \partial ^{\mu}+m^2)\Phi = 0 $</WRAP> |

+ | ====== Klein-Gordon Equation ====== | ||

- | <note tip> $$( \partial _{\mu} \partial ^{\mu}+m^2)\Phi = 0 $$ </note> | + | <tabbox Intuitive> |

- | <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 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|]]. | ||

- | <tabbox Layman> | + | 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 25: | 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.1520936150.txt.gz · Last modified: 2018/03/13 10:15 (external edit)

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