equations:klein-gordon_equation
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equations:klein-gordon_equation [2018/03/13 11:15] jakobadmin |
equations:klein-gordon_equation [2025/03/04 00:45] (current) edi [Concrete] |
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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/appendices/#field_equations|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.1520936137.txt.gz · Last modified: 2018/03/13 10:15 (external edit)
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