|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{Quantum mechanics|cTopic=Equations}}
| |
| The '''Klein–Gordon equation''' ('''Klein–Fock–Gordon equation''' or sometimes '''Klein–Gordon–Fock equation''') is a [[special relativity|relativistic]] version of the [[Schrödinger equation]].
| |
|
| |
|
| It is the equation of motion of a [[quantum field theory|quantum scalar or pseudoscalar field]], a field whose quanta are spinless particles. It cannot be straightforwardly interpreted as a [[Schrödinger equation]] for a quantum state, because it is second order in time and because it does not admit a positive definite conserved probability density. Still, with the appropriate [[Feynman-Stueckelberg interpretation#The_Feynman-Stueckelberg_interpretation|interpretation]], it does describe the quantum amplitude for finding a point particle in various places, the relativistic wavefunction, but the particle propagates both forwards and backwards in time. Any solution to the [[Dirac equation]] is automatically a solution to the Klein–Gordon equation, but the converse is not true.
| |
|
| |
|
| ==Statement==
| | Regardless of the Clash of Clans hack tool; there include also hack tools for other games. Those can check out those hacks and obtain hundreds of which they need. It is sure that will have lost at fun once they keep the [http://www.google.com/search?q=hack+tool&btnI=lucky hack tool] that they can.<br><br>If you loved this article and you simply would like to [http://search.un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=receive&Submit=Go receive] more info concerning [http://prometeu.net clash of clans generator] nicely visit the internet site. But Supercell, by allowing those illusion on the multi player game, taps into those instinctual male drive towards from the status hierarchy, and even though it''s unattainable to the top of your hierarchy if there's no need been logging in every single because the game arrived on the scene plus you invested fundamental money in extra builders, the drive for obtaining a small bit further compels enough visitors to spend a real income with regards to virtual 'gems'" that game could be the top-grossing app within the Mobile application Store.<br><br>Pay attention to a game's evaluation when purchasing an existing. This evaluation will allow you recognize what age level clash of clans hack tool is right for and will reveal to you when the sport is violent. It figure out whether you should buy the sport.<br><br>Guilds and clans have already been popular ever since the primary beginning of first-person supply shooter and MMORPG game playing. World of WarCraft develops fot it concept with their very own World associated Warcraft guilds. A real guild can easily always remain understood as a having to do with players that band down for companionship. Individuals the guild travel together again for fun and adventure while improving in experience and gold.<br><br>You might be looking Conflict of Their families Jewels Free, or you should be just buying a Skimp Conflict of Tribes, possess the smartest choice on the internet, absolutely free as well as only takes a couple of minutes to get all these products.<br><br>A person's war abject is agnate in your approved village, except that your showdown abject will not accomplish resources. Barrio inside your warfare abject will not be anon improved per rearranged, as it isolated mimics this adjustment and then accomplished completed advancement degrees of your apple inside alertness day. Combat bases additionally never fees to take their goodies rearmed, defenses reloaded in addition to characters healed, as they are consistently ready. The association alcazar around the your war abject service charge be abounding alone concerned with the one in ones own whole village.<br><br>Let's try interpreting the proper abstracts differently. Believe of it in agreement of bulk with gems to skip 1 subsequently. Skipping added moments expenses added money, and you get a larger motors deal. Think to do with it as a not many accretion discounts. |
| The Klein–Gordon equation is
| |
| :<math> \frac {1}{c^2} \frac{\partial^2}{\partial t^2} \psi - \nabla^2 \psi + \frac {m^2 c^2}{\hbar^2} \psi = 0. </math> | |
| | |
| This is often abbreviated as
| |
| :<math>(\Box + \mu^2) \psi = 0,</math>
| |
| | |
| where <math> \mu = \dfrac{mc}{\hbar}</math> and <math>\Box</math> is the [[d'Alembert operator]], defined by
| |
| | |
| :<math> \Box = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2.</math> | |
| | |
| The equation is most often written in [[natural units]]:
| |
| :<math> - \partial_t^2 \psi + \nabla^2 \psi = m^2 \psi</math>
| |
| | |
| The form is determined by requiring that [[plane wave]] solutions of the equation:
| |
| :<math>\psi = e^{-i\omega t + i k\cdot x } = e^{i k_\mu x^\mu}</math>
| |
| | |
| obey the energy momentum relation of special relativity:
| |
| :<math> -p_\mu p^\mu = E^2 - P^2 = \omega^2 - k^2 = - k_\mu k^\mu = m^2</math>
| |
| | |
| Unlike the Schrödinger equation, the Klein–Gordon equation admits two values of ω for each ''k'', one positive and one negative. Only by separating out the positive and negative frequency parts does one obtain an equation describing a relativistic wavefunction. For the time-independent case, the Klein–Gordon equation becomes
| |
| | |
| :<math>\left[ \nabla^2 - \frac {m^2 c^2}{\hbar^2} \right] \psi(\mathbf{r}) = 0</math>
| |
| | |
| which is the homogeneous [[screened Poisson equation]].
| |
| | |
| ==History== | |
| The equation was named after the physicists [[Oskar Klein]] and [[Walter Gordon (physicist)|Walter Gordon]], who in 1926 proposed that it describes relativistic electrons. Other authors making similar claims in that same year were [[Vladimir Fock]], [[Johann Kudar]], [[Théophile de Donder]] and [[Frans-H. van den Dungen]], and [[Louis de Broglie]]. Although it turned out that the [[Dirac equation]] describes the spinning electron, the Klein–Gordon equation correctly describes the spinless [[pion]], a [[composite particle]]. On July 4th, 2012 [[CERN]] announced the discovery of the first spin-zero elementary particle, the [[Higgs boson]]. Further experimentation and analysis is required to discern if the [[Higgs Boson]] found is that of the [[Standard Model]], or a more exotic form.
| |
| | |
| The Klein–Gordon equation was first considered as a quantum wave equation by [[Erwin Schrödinger|Schrödinger]] in his search for an equation describing de Broglie waves. The equation is found in his notebooks from late 1925, and he appears to have prepared a manuscript applying it to the hydrogen atom. Yet, because it fails to take into account the electron's spin, the equation predicts the hydrogen atom's fine structure incorrectly, including overestimating the overall magnitude of the splitting pattern by a factor of <math>\tfrac{4n}{2n-1}</math> for the ''n''-th energy level. The Dirac result is, however, easily recovered if the orbital momentum quantum number ''l'' is replaced by total angular momentum quantum number ''j''.<ref name="Itzykson&Zuber">See C Itzykson and J-B Zuber, Quantum Field Theory, McGraw-Hill Co., 1985, pp. 73-74. Eq. 2.87 is identical to eq. 2.86 except that it features ''j'' instead of ''l''.</ref> In January 1926, Schrödinger submitted for publication instead ''his'' equation, a non-relativistic approximation that predicts the Bohr energy levels of hydrogen without fine structure.
| |
| | |
| In 1927, soon after the Schrödinger equation was introduced, [[Vladimir Aleksandrovich Fock|Vladimir Fock]] wrote an article about its generalization for the case of [[magnetic field]]s, where [[force]]s were dependent on [[velocity]], and independently derived this equation. Both Klein and Fock used Kaluza and Klein's method. Fock also determined the [[gauge theory]] for the [[wave equation]]. The Klein–Gordon equation for a [[free particle]] has a simple [[plane wave]] solution.
| |
| | |
| ==Derivation==
| |
| The non-relativistic equation for the energy of a free particle is
| |
| :<math>\frac{\mathbf{p}^2}{2 m} = E.</math>
| |
| | |
| By quantizing this, we get the non-relativistic Schrödinger equation for a free particle,
| |
| :<math>\frac{\mathbf{\hat{p}}^2}{2m} \psi = \hat{E}\psi</math>
| |
| where
| |
| :<math>\mathbf{\hat{p}} =-i \hbar \mathbf{\nabla}</math>
| |
| | |
| is the [[momentum operator]] (<math>\mathbf\nabla</math> being the [[del|del operator]]), and
| |
| :<math>\hat{E}=i \hbar \dfrac{\partial}{\partial t}</math>
| |
| | |
| is the [[energy operator]].
| |
| | |
| The Schrödinger equation suffers from not being relativistically covariant, meaning it does not take into account Einstein's [[special relativity]].
| |
| | |
| It is natural to try to use the identity from special relativity describing the energy:
| |
| | |
| :<math>\sqrt{\mathbf{p}^2 c^2 + m^2 c^4} = E</math>
| |
| | |
| Then, just inserting the quantum mechanical operators for momentum and energy yields the equation
| |
| | |
| :<math> \sqrt{(-i\hbar\mathbf{\nabla})^2 c^2 + m^2 c^4} \psi = i \hbar \frac{\partial}{\partial t}\psi. </math>
| |
| | |
| This, however, is a cumbersome expression to work with because the differential operator cannot be evaluated while under the square root sign. In addition, this equation, as it stands, is [[Nonlocality|nonlocal]] (see also [http://www.ma.utexas.edu/mediawiki/index.php/Introduction_to_nonlocal_equations Introduction to nonlocal equations]).
| |
| | |
| Klein and Gordon instead began with the square of the above identity, i.e.
| |
| :<math>\mathbf{p}^2 c^2 + m^2 c^4 = E^2</math>
| |
| | |
| which, when quantized, gives
| |
| :<math> \left ((-i\hbar\mathbf{\nabla})^2 c^2 + m^2 c^4 \right ) \psi = \left(i \hbar \frac{\partial}{\partial t} \right)^2 \psi </math>
| |
| | |
| which simplifies to
| |
| :<math> - \hbar^2 c^2 \mathbf{\nabla}^2 \psi + m^2 c^4 \psi = - \hbar^2 \frac{\partial^2}{(\partial t)^2} \psi. </math>
| |
| | |
| Rearranging terms yields
| |
| :<math> \frac {1}{c^2} \frac{\partial^2}{(\partial t)^2} \psi - \mathbf{\nabla}^2 \psi + \frac {m^2 c^2}{\hbar^2} \psi = 0. </math>
| |
| | |
| Since all reference to imaginary numbers has been eliminated from this equation, it can be applied to fields that are [[real number|real]] valued as well as those that have [[complex number|complex]] values.
| |
| | |
| Using the inverse of the [[Minkowski metric]] diag(−''c''<sup>2</sup>, 1, 1, 1), we get
| |
| :<math> - \eta^{\mu \nu} \partial_{\mu} \partial_{\nu} \psi + \frac {m^2 c^2}{\hbar^2} \psi = 0</math>
| |
| | |
| in [[Lorentz covariant|covariant]] notation. This is often abbreviated as
| |
| :<math>(\Box + \mu^2) \psi = 0,</math>
| |
| | |
| where
| |
| | |
| :<math> \mu = \frac{mc}{\hbar}</math>
| |
| | |
| and | |
| | |
| :<math> \Box = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2.</math>
| |
| | |
| This operator is called the [[d'Alembert operator]]. Today this form is interpreted as the relativistic [[field equation]] for a scalar (i.e. [[spin (physics)|spin]]-0) particle. Furthermore, any solution to the [[Dirac equation]] (for a spin-one-half particle) is automatically a solution to the Klein–Gordon equation, though not all solutions of the Klein–Gordon equation are solutions of the Dirac equation. It is noteworthy that the Klein–Gordon equation is very similar to the [[Proca action|Proca equation]].
| |
| | |
| === Klein-Gordon equation in a potential ===
| |
| | |
| The Klein–Gordon equation can be generalized to describe a field in some potential ''V''(''ψ'') as:<ref>David Tong, [http://www.damtp.cam.ac.uk/user/tong/qft.html Lectures on Quantum Field Theory], Lecture 1, Section 1.1.1</ref>
| |
| | |
| :<math>\Box \psi + \frac{\partial{}V}{\partial \psi} = 0</math>
| |
| | |
| == Relativistic free particle solution ==
| |
| | |
| The Klein–Gordon equation for a free particle can be written as
| |
| | |
| :<math>\mathbf{\nabla}^2\psi-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\psi = \frac{m^2c^2}{\hbar^2}\psi</math>
| |
| | |
| with the same solution as in the non-relativistic case: | |
| | |
| :<math>\psi(\mathbf{r}, t) = e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}</math>
| |
| | |
| except with the constraint, known as the dispersion relation:
| |
| | |
| :<math>-k^2+\frac{\omega^2}{c^2}=\frac{m^2c^2}{\hbar^2}.</math>
| |
| | |
| Just as with the non-relativistic particle, we have for energy and momentum:
| |
| | |
| :<math>\langle\mathbf{p}\rangle=\left\langle \psi \left|-i\hbar\mathbf{\nabla}\right|\psi\right\rangle = \hbar\mathbf{k},</math>
| |
| :<math>\langle E\rangle=\left\langle \psi \left|i\hbar\frac{\partial}{\partial t}\right|\psi\right\rangle = \hbar\omega.</math>
| |
| | |
| Except that now when we solve for k and ω and substitute into the constraint equation, we recover the relationship between energy and momentum for relativistic massive particles:
| |
| | |
| :<math>\langle E \rangle^2=m^2c^4+\langle \mathbf{p} \rangle^2c^2.</math>
| |
| | |
| For massless particles, we may set ''m'' = 0 in the above equations. We then recover the relationship between energy and momentum for massless particles:
| |
| | |
| :<math>\langle E \rangle=\langle |\mathbf{p}| \rangle c.</math>
| |
| | |
| ==Action==
| |
| The Klein–Gordon equation can also be derived via a [[calculus of variations|variational]] method by considering the action:
| |
| | |
| :<math>\mathcal{S} = \int \left( - \frac{\hbar^2}{m} \eta^{\mu \nu} \partial_{\mu}\bar\psi \partial_{\nu}\psi - m c^2 \bar\psi \psi \right) \mathrm{d}^4 x </math>
| |
| | |
| where <math>\psi </math> is the Klein–Gordon field and ''m'' is its mass. The [[complex conjugate]] of <math>\psi </math> is written <math>\bar\psi .</math> If the scalar field is taken to be real-valued, then <math>\bar\psi = \psi .</math>
| |
| | |
| Applying the formula for the [[stress-energy tensor#Hilbert stress–energy tensor|Hilbert stress–energy tensor]] to the Lagrangian (the quantity inside the integral), we can derive the [[stress-energy tensor]] of the scalar field. It is
| |
| :<math>T^{\mu\nu} = \frac{\hbar^2}{m} \left (\eta^{\mu \alpha} \eta^{\nu \beta} + \eta^{\mu \beta} \eta^{\nu \alpha} - \eta^{\mu\nu} \eta^{\alpha \beta} \right ) \partial_{\alpha}\bar\psi \partial_{\beta}\psi - \eta^{\mu\nu} m c^2 \bar\psi \psi .</math>
| |
| | |
| ==Electromagnetic interaction==
| |
| There is a simple way to make any field interact with electromagnetism in a [[gauge theory|gauge invariant]] way: replace the derivative operators with the gauge covariant derivative operators. The Klein Gordon equation becomes:
| |
| | |
| :<math>D_\mu D^\mu \phi = -(\partial_t - ie A_0)^2 \phi + (\partial_i - ie A_i)^2 \phi = m^2 \phi</math>
| |
| | |
| in [[natural units]], where A is the vector potential. While it is possible to add many higher order terms, for example,
| |
| | |
| :<math>D_\mu D^\mu\phi + A F^{\mu\nu} D_\mu \phi D_\nu (D_\alpha D^\alpha \phi) =0</math>
| |
| these terms are not [[renormalization|renormalizable]] in 3+1 dimensions.
| |
| | |
| The field equation for a charged scalar field multiplies by i, which means the field must be complex. In order for a field to be charged, it must have two components that can rotate into each other, the real and imaginary parts.
| |
| | |
| The action for a charged scalar is the covariant version of the uncharged action: | |
| | |
| :<math>S= \int_x \left (\partial_\mu \phi^* + ie A_\mu \phi^* \right ) \left (\partial_\nu \phi - ie A_\nu\phi \right )\eta^{\mu\nu} = \int_x |D \phi|^2</math>
| |
| | |
| ==Gravitational interaction==
| |
| In [[general relativity]], we include the effect of gravity and the Klein–Gordon equation becomes<ref>S.A. Fulling, Aspects of Quantum Field Theory in Curved Space-Time, Cambridge University Press, 1996, p. 117</ref>
| |
| | |
| :<math>\begin{align}
| |
| 0 & = - g^{\mu \nu} \nabla_{\mu} \nabla_{\nu} \psi + \dfrac {m^2 c^2}{\hbar^2} \psi = - g^{\mu \nu} \nabla_{\mu} (\partial_{\nu} \psi) + \dfrac {m^2 c^2}{\hbar^2} \psi \\
| |
| & = - g^{\mu \nu} \partial_{\mu} \partial_{\nu} \psi + g^{\mu \nu} \Gamma^{\sigma}{}_{\mu \nu} \partial_{\sigma} \psi + \dfrac {m^2 c^2}{\hbar^2} \psi
| |
| \end{align}</math>
| |
| | |
| or equivalently
| |
| | |
| :<math>\frac{-1}{\sqrt{-g}} \partial_{\mu} \left ( g^{\mu \nu} \sqrt{-g} \partial_{\nu} \psi \right ) + \frac {m^2 c^2}{\hbar^2} \psi = 0</math>
| |
| | |
| where <math>g^{\alpha \beta}</math> is the inverse of the [[metric tensor]] that is the gravitational potential field, ''g'' is the [[determinant]] of the metric tensor,
| |
| <math>\nabla_{\mu}</math> is the [[covariant derivative]] and <math>\Gamma^{\sigma}{}_{\mu \nu}</math> is the [[Christoffel symbol]] that is the gravitational [[force field (physics)|force field]].
| |
| | |
| ==Traveling wave solution==
| |
| There exists traveling wave solutions in the form of solitons for Klein-Gordon equation
| |
| <ref>Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple p64-72 Springer</ref>
| |
| {{Gallery
| |
| |width=250
| |
| |height=200
| |
| |align=center
| |
| |File:Klein Gordon equation traveling wave plot4.gif|Klein Gordon equation traveling wave plot4
| |
| |File:Klein Gordon equation traveling wave plot5.gif|Klein Gordon equation traveling wave plot5
| |
| |File:Klein Gordon equation traveling wave plot6.gif|Klein Gordon equation traveling wave plot6
| |
| }}
| |
| | |
| | |
| | |
| ==See also==
| |
| *[[Dirac equation]]
| |
| *[[Rarita–Schwinger equation]]
| |
| *[[Quantum field theory]]
| |
| *[[Scalar field theory]]
| |
| *[[Sine–Gordon equation]]
| |
| | |
| ==Notes==
| |
| <references/>
| |
| | |
| ==References==
| |
| *{{cite book | author=Sakurai, J. J. | title=Advanced Quantum Mechanics | publisher=Addison Wesley | year=1967 | isbn=0-201-06710-2}}
| |
| *{{cite book | author= Davydov, A.S. | title= Quantum Mechanics, 2nd Edition | publisher=Pergamon | year=1976 | isbn=0-08-020437-6}}
| |
| | |
| ==External links==
| |
| * {{springer|title=Klein-Gordon equation|id=p/k055480}}
| |
| * {{MathWorld| urlname=Klein-GordonEquation | urltitle=Klein–Gordon equation}}
| |
| * [http://eqworld.ipmnet.ru/en/solutions/lpde/lpde203.pdf Linear Klein–Gordon Equation] at EqWorld: The World of Mathematical Equations.
| |
| * [http://eqworld.ipmnet.ru/en/solutions/npde/npde2107.pdf Nonlinear Klein–Gordon Equation] at EqWorld: The World of Mathematical Equations.
| |
| * [http://www.ma.utexas.edu/mediawiki/index.php/Introduction_to_nonlocal_equations Introduction to nonlocal equations].
| |
| | |
| {{DEFAULTSORT:Klein-Gordon equation}}
| |
| [[Category:Concepts in physics]]
| |
| [[Category:Partial differential equations]]
| |
| [[Category:Equations]]
| |
| [[Category:Special relativity]]
| |
| [[Category:Waves]]
| |
| [[Category:Quantum field theory]]
| |