# 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.

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9. ### A Seeking a derivation of Schrödinger's wave equation

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...
10. ### Dodelson Cosmology 6.8 Inflation Klein Gordon Equation

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...
11. ### How to show speed is equal to group velocity?

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...
12. ### I Is the ground state energy of a quantum field actually zero?

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...

23. ### The use in solving the Klein Gordon equation?

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...
24. ### Klein-Gordon Hamiltonian commutator

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...
25. ### Klein-Gordon eqn: why dismiss messages at phase velocity

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...
26. ### How does one derive the Lagrangian densities used in QFT?

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...
27. ### Klein Gordon Equation in Quintessence Models

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...
28. ### Solutions of the free one-particle Klein Gordon equation

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...
29. ### Time Dependent Perturbation Theory - Klein Gordon Equation

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 +...
30. ### Klein Gordon equation and particles with spin

I am a newbie to QM. Why can't the Klein Gordon equation be used to describe particles with spin? Thanks
31. ### Solution to the Klein Gordon Equation

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...
32. ### The Lorentz force derived from the Klein Gordon equation

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...
33. ### Solving the Massless Klein Gordon Equation

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...
34. ### Klein Gordon equation, probability density

[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})...
35. ### Klein-Gordon vs Schrodinger-Fock Equation

hi all.................are d klein gordon equation n d schrodinger fock equation d same?