What is Klein gordon equation: Definition and 35 Discussions
The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. It is a quantized version of the relativistic energy–momentum relation. Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. Its theoretical relevance is similar to that of the Dirac equation. Electromagnetic interactions can be incorporated, forming the topic of scalar electrodynamics, but because common spinless particles like the pions are unstable and also experience the strong interaction (with unknown interaction term in the Hamiltonian,) the practical utility is limited.
The equation can be put into the form of a Schrödinger equation. In this form it is expressed as two coupled differential equations, each of first order in time. The solutions have two components, reflecting the charge degree of freedom in relativity. It admits a conserved quantity, but this is not positive definite. The wave function cannot therefore be interpreted as a probability amplitude. The conserved quantity is instead interpreted as electric charge, and the norm squared of the wave function is interpreted as a charge density. The equation describes all spinless particles with positive, negative, and zero charge.
Any solution of the free Dirac equation is, for each of its four components, a solution of the free Klein–Gordon equation. The Klein–Gordon equation does not form the basis of a consistent quantum relativistic one-particle theory. There is no known such theory for particles of any spin. For full reconciliation of quantum mechanics with special relativity, quantum field theory is needed, in which the Klein–Gordon equation reemerges as the equation obeyed by the components of all free quantum fields. In quantum field theory, the solutions of the free (noninteracting) versions of the original equations still play a role. They are needed to build the Hilbert space (Fock space) and to express quantum fields by using complete sets (spanning sets of Hilbert space) of wave functions.
Hello everyone. The 1D wave equation is written:
$$ \left( \partial_t^2/c^2 - \partial_x^2 \right) \Psi = 0$$
An electromagnetic wave or matter wave, like free electron (unnormalized here), can be written with the following wave function ##\Psi_m## of energy ## \hbar k c ##:
$$ \Psi_m \propto...
At non-relativistic limit, m>>p so let p=0
At non-relativistic limit m>>w,
So factorise out m^2 from the square root to get:
m*sqrt(1+2w(n+1/2)/m)
Taylor expansion identity for sqrt(1+x) for small x gives:
E=m+w(n+1/2) but it should equal E=p^2/2m +w(n+1/2), so how does m transform into p^2/2m?
I am trying to numerically solve (with Mathematica) a relativistic version of infinite square well with an oscillating wall using Klein-Gordon equation. Firstly, I transform my spatial coordinate ## x \to y = \frac{x}{L[t]} ## to make the wall look static (this transformation is used a lot in...
Given that the Minkowski metric implies the Lorentz transformations and special relativity, why do the equations of relativistic quantum mechanics, i.e., the Dirac and Klein-Gordon equations, require a mass term to unite quantum mechanics and special relativity? Shouldn't their formulation in...
The Klein-Gordon equation is based on the relation
(E-eΦ)2-(pc-eA)2=m2^2c2, which is the magnitude of the difference between the momentum four-vector and the four-potential.
Since the magnitude of the momentum four-vector is given by
E2-p2c2=m2c4, does it follow that the magnitude of the...
Hey everybody,
Background:
I'm currently working on a toy model for my master thesis, the massless Klein-Gordon equation in a rotating static Kerr-Schild metric.
The partial differential equations are (see http://arxiv.org/abs/1705.01071, equation 27, with V'=0):
$$ \partial_t\phi =...
I am interested in the derivation of Schrödinger’s wave equation from the Klein Gordon equation. I have looked in Penfold’s ‘The Road to Reality’, the open University’s Quantum Mechanics books, Feynman’s lectures, the internet, but not found what I want. Everyone seems to take it as a given...
Homework Statement
Show that Eq. (6.33) follows from Eq. (6.32) by changing variables from t to ##\eta##.
Homework Equations
(6.32) $$\frac{d^2\phi^{(0)}}{dt^2}+3H\frac{d\phi^{(0)}}{dt}+V'=0$$
(6.33) $$\ddot{\phi^{(0)}}+2aH\dot{\phi}^{(0)}+a^2V'=0$$
The Attempt at a Solution
So...
Homework Statement
My question is, how do I show that speed is equal to group velocity? More information at https://imgur.com/a/m6FwNaG
Homework Equations
v_g = dw/dk
The Attempt at a Solution
Part a is substitution, part b uses v_g = dw/dk, part c is multiplication by h-bar, but I am stuck...
I start by outlining the little I know about the basics of quantum field theory.
The simplest relativistic field theory is described by the Klein-Gordon equation of motion for a scalar field ##\large \phi(\vec{x},t)##:
$$\large \frac{\partial^2\phi}{\partial t^2}-\nabla^2\phi+m^2\phi=0.$$
We...
Homework Statement
I am considering the Klein Gordon Equation in a box with Dirichlet conditions (i.e., ##\hat{\phi}(x,t)|_{boundary} = 0 ##). 1-D functions that obey the Dirichlet condition on interval ##[0,L]## are of the form below (using the discrete Fourier sine transform)
$$f(x) =...
Hi everyone,
I've been reading about the Klein Gordon equation with the Coulomb Potential. The full solution can be found here:
http://wiki.physics.fsu.edu/wiki/index.php/Klein-Gordon_equation#Klein-Gordon_equation_with_Coulomb_potential
I'm confused near the beginning of this. I understand...
If we have the normal KG scalar field expansion:
$$ \hat{\phi}(x^{\mu}) = \int \frac{d^{3}p}{(2\pi)^{3}\omega(\mathbf{p})} \big( \hat{a}(p)e^{-ip_{\mu}x^{\mu}}+\hat{a}^{\dagger}(p)e^{ip_{\mu}x^{\mu}} \big) $$
With ## \omega(\mathbf{p}) = \sqrt{|\mathbf{p}^{2}|+m^{2}}##
Then why do we associate...
So, I am a newbie in quantum mechanics, took modern physics last fall for my physics minor.
I know that Schrodinger based his equation based on the equation K + V = E,
by using non-relativistic kinematic energy (P2/2m + V = E)
p becoming the operator p= -iħ∇ for the wave equation eigenfunction...
Suppose one starts with the standard Klein-Gordon (KG) Lagrangian for a free scalar field: $$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}$$ Integrating by parts one can obtain an equivalent (i.e. gives the same equations of motion) Lagrangian...
I'm studying Quantum Field Theory and the first example being given in the textbook is the massless Klein Gordon field whose equation is just the wave equation \Box \ \phi = 0. The only problem is that I'm not being able to get the same solution as the book. In the book the author states that...
Consider the double-slit experiment done with photons from a laser. If one was interested only in computing position (vertical) probability amplitudes and did not care about spin/helicity, could the Klein-Gordon Equation (with mass set to zero) be used?
Thanks in advance.
If equation of motion(K-G Eqn.,) follows,
∂μ∂μΦ+m2Φ=ρ
where 'ρ' is point source at origin.
How time independent form of above will become,
(∇2-m2)Φ(x)=gδ3(x)
where g is the coupling constant,
δ3(x) is three dimensional dirac delta function.
Homework Statement
Consider the Klein-Gordon equation ##(\partial_\mu \partial^{\mu}+m^2)\varphi(x)=0##. Using Noether's theorem, find a continuity equation of the form ##\partial_\mu j^{\mu}=0##.
Homework Equations
##(\partial_\mu \partial^{\mu}+m^2)\varphi(x)=0##
The Attempt at a Solution...
I understand that the ansatz to $$(\Box +m^{2})\phi(\mathbf{x},t)=0$$ (where ##\Box\equiv\partial^{\mu}\partial_{\mu}=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}##) is of the form ##\phi(\mathbf{x},t)=e^{(iE_{\mathbf{k}}t-\mathbf{k}\cdot\mathbf{x})}##, where...
I've done some reading on quantum field theory, and I went over how when Schrodinger first derived this equation, he discarded because it yielded negative energy solutions, negative probability distributions and it gave an incorrect spectrum for the hydrogen atom. The book then went on to state...
Homework Statement
Consider the quantum mechanical Hamiltonian ##H##. Using the commutation relations of the fields and conjugate momenta , show that if ##F## is a polynomial of the fields##\Phi## and ##\Pi## then
##[H,F]-i \partial_0 F##
Homework Equations
For KG we have:
##H=\frac{1}{2} \int...
Hi All,
I've heard it said that the superluminal phase velocity of the KG eqn is not a problem for relativistic causality because signals travel at the packet/group velocity, which is the inverse of the phase velocity (c being 1). I'm a bit skeptical of this.
We can strip away all the quantum...
I've been working through a qft book by Sadovskii (while I wait for my Peskin book to come in) and I've used some later chapters of Griffith's Into to Elementary Particles as an introduction to some qft. My issue with both of these is that, where in classical mechanics we have the Lagrangian...
Hello! I'm studying various dark energy models, and as a part of the project, I need to be able to numerically solve the Klein-Gordon (KG) equation and the Friedmann Equation (FE) in the context of a canonical scalar field. I wasn't sure whether or not this belonged here or in the computational...
In the book "Wachter, relativistic quantum mechanics", in page 5, the KG eq. is introduced as follows:
-\hbar^2 \frac{\partial^2 \phi(x)}{\partial t^2} = (-c^2 \hbar^2 \nabla^2 + m^2_0 c^4) \phi(x).
Now I tried to solve this equation using the separation ansatz (product ansatz).
I get...
Hey,
I'm struggling to understand a number of things to do with this derivation of the scattering amplitude using time dependent perturbation theory for spinless particles.
We assume we have some perturbation 'V' such that :
\left ( \frac{\partial^2 }{\partial t^2}-\triangledown ^2 +...
Hey guys, I was reading up on the Klein Gordon equation and I came across an article that gave a general solution as: \psi(r,t)= e^i(kr-\omegat), under the constraint that -k^2 + \omega^2/c^2 = m^2c^2/\hbar^2, forgive my lack of latex hah.
Through Euler's law this does give a solution...
The Lorentz Force and Maxwell's equations derived from Klein Gordon's equation
.
http://www.physics-quest.org/Book_Lorentz_force_from_Klein_Gordon.pdfI posted several new chapters of my book lately, mostly involving the Klein Gordon equation.
This chapter shows how the Lorentz Force has a...
I am being really thick here
I have this wave equation, the massless klien gordon equation
\partial_{\mu}\partial^{\mu}\phi(x)=0
where the summation over \mu is over 0,1,2,3
the general solution is a superposition of plane waves yes? i.e
\phi(x)=\int d^4 p...
[SOLVED] Klein Gordon equation, probability density
Homework Statement
Use the Klein-Gordon Equation to show that
\partial_{\mu}j^{\mu} = 0
Homework Equations
KG:
\left(\frac{\partial^{2}}{\partial t^{2}} - \nabla^{2} + m^{2}\right) \phi = (\partial_{\mu}\partial^{\mu} + m^{2})...