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 equations:klein-gordon_equation [2018/03/28 10:23]jakobadmin equations:klein-gordon_equation [2021/03/31 18:22] (current)edi [Concrete] Both sides previous revision Previous revision 2021/03/31 18:22 edi [Concrete] 2019/07/30 08:52 [Why is it interesting?] 2019/07/30 08:46 [Klein-Gordon Equation] 2019/07/30 08:32 [Klein-Gordon Equation] 2018/04/16 09:12 jakobadmin [Intuitive] 2018/03/29 13:56 leot221 [Concrete] 2018/03/29 13:55 leot221 [Concrete] 2018/03/28 10:23 jakobadmin 2018/03/26 16:40 jakobadmin 2018/03/26 16:39 jakobadmin 2018/03/13 11:25 jakobadmin 2018/03/13 11:18 jakobadmin 2018/03/13 11:18 jakobadmin 2018/03/13 11:18 jakobadmin 2018/03/13 11:18 jakobadmin 2018/03/13 11:15 jakobadmin 2018/03/13 11:15 jakobadmin 2018/03/13 11:14 jakobadmin [Why is it interesting?] 2017/12/04 08:01 external edit2017/11/05 18:00 [Student] 2017/10/21 15:23 jakobadmin [Student] 2017/10/21 15:22 jakobadmin [Why is it interesting?] 2017/10/21 15:21 jakobadmin created Next revision Previous revision 2021/03/31 18:22 edi [Concrete] 2019/07/30 08:52 [Why is it interesting?] 2019/07/30 08:46 [Klein-Gordon Equation] 2019/07/30 08:32 [Klein-Gordon Equation] 2018/04/16 09:12 jakobadmin [Intuitive] 2018/03/29 13:56 leot221 [Concrete] 2018/03/29 13:55 leot221 [Concrete] 2018/03/28 10:23 jakobadmin 2018/03/26 16:40 jakobadmin 2018/03/26 16:39 jakobadmin 2018/03/13 11:25 jakobadmin 2018/03/13 11:18 jakobadmin 2018/03/13 11:18 jakobadmin 2018/03/13 11:18 jakobadmin 2018/03/13 11:18 jakobadmin 2018/03/13 11:15 jakobadmin 2018/03/13 11:15 jakobadmin 2018/03/13 11:14 jakobadmin [Why is it interesting?] 2017/12/04 08:01 external edit2017/11/05 18:00 [Student] 2017/10/21 15:23 jakobadmin [Student] 2017/10/21 15:22 jakobadmin [Why is it interesting?] 2017/10/21 15:21 jakobadmin created Line 5: Line 5:  ​  ​ - + 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 formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. + - ​ + 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]]. ​ ​  ​  ​ + The Klein-Gordon equation can be derived from the Lagrangian + + $$\mathscr{L}= \frac{1}{2}( \partial _{\mu} \Phi \partial ^{\mu} \Phi -m^2 \Phi^2)$$ + + using the [[equations:​euler_lagrange_equations|Euler-Lagrange equations]]. + + ---- + + **Solutions** + + The most general solution of the Klein-Gordon equation is$$\label{KGsol} \Phi(x)= \int \mathrm{d }k^3 \frac{1}{(2\pi)^3 2\omega_k} \left( a(k){\mathrm{e }}^{ -i(k x)} + a^\dagger(k) {\mathrm{e }}^{ i(kx)}\right) .$$ + + ---- * 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}}  ​  ​ Line 22: Line 46:  ​  ​ - 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. ​ ​ Line 30: Line 54: * $\partial _{\mu}$ 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: $\Phi$ 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.