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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  1. F

    I Non-wave solution to wave equation and virtual particles

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  2. D

    Landau Energy Spectrum in the non-relativistic limit

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  3. F

    I Infinite Square Well with an Oscillating Wall (Klein-Gordon Equation)

    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...
  4. J

    A Spin-One Klein Gordon Equation

    What is the spin one Klein Gordon Equation? What is the formula for the conserved current, i.e. the electric current density four-vector?
  5. koustav

    A Klein-Gordon Equation: Solving 2nd Order Time Derivative

    What problem actually arises when we take the second order time derivative in KG equation
  6. redtree

    I Relativistic quantum mechanics

    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...
  7. T

    A Magnitude 4-Vector Lorenz Gauge: Klein-Gordon Eq.

    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...
  8. T

    A Numerically Solving Scalar Propagation in Curved Spacetime

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  9. G

    A Seeking a derivation of Schrödinger's wave equation

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  10. S

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  11. Y

    How to show speed is equal to group velocity?

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  12. J

    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...
  13. D

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  14. xhwvnsghsfasd

    A Coulomb Klein Gordon: Where does e^(-iEt) come from?

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  15. bananabandana

    I Negative and Positive energy modes of KG equation

    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...
  16. C

    I Deductions of Formulas for Energy

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  17. F

    I Equivalent Klein-Gordon Lagrangians and equations of motion

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  18. L

    I Wave equation solution using Fourier Transform

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  19. LarryS

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  20. M

    A Time Independent Form of Klein Gordon Eqn.: How to Reach (gδ3(x))

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  21. It's me

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  22. D

    A How to derive general solution to the Klein-Gordon equation

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  23. S

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  24. loops496

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  25. A

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  26. M

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  27. T

    Klein Gordon Equation in Quintessence Models

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  28. S

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  29. S

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  30. N

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  31. B

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  32. Hans de Vries

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  33. B

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  34. O

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  35. A

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