Klein-Gordon equation

The one-dimensional Klein-Gordon equation

ϕ_{tt} - c^{2} ϕ_{xx} + A^{2} c^{2} ϕ = 0

has positive-frequency dispersion relation

ω (k) = c k^{2} + A^{2}

.

Table of contents
- Stationary-phase asymptotic of a Klein-Gordon wave along a subluminal ray Klein-Gordon equation
  - Upward zero crossings of an oscillatory stationary-phase tail Stationary-phase asymptotic of a Klein-Gordon wave along a subluminal ray

Stationary-phase asymptotic of a Klein-Gordon wave along a subluminal ray

For zero initial velocity and Fourier amplitude

a (k)

, observation along

x = V t

with

0 \leq V < c

selects the two stationary wavenumbers

\pm k_{0}

, where

k_{0} = \frac{A V}{c ^{2} - V ^{2}} .

s = c^{2} - V^{2}

, the leading oscillation has angular frequency

A s

and amplitude proportional to

t^{- 1/2}

.

Upward zero crossings of an oscillatory stationary-phase tail

If the leading asymptotic is

C t^{- 1/2} cos (Ω t + γ)

with

C > 0

, its upward zero crossings satisfy

Ω t + γ = \frac{3 π}{2} + 2 πn

to leading order.

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Klein–Gordon equation by

Wikipedia Bot  1

The Klein-Gordon equation is a relativistic wave equation for scalar particles, derived from both quantum mechanics and special relativity. It describes the dynamics of a scalar field, which represents a particle of spin-0 (such as a pion or any other fundamental scalar particle).

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Klein-Gordon equation by

Ciro Santilli  40 Updated 2025-07-16

A relativistic version of the Schrödinger equation.

Correctly describes spin 0 particles.

The most memorable version of the equation can be written as shown at Section "Klein-Gordon equation in Einstein notation" with Einstein notation and Planck units:

\partial_{i} \partial^{i} ψ - m^{2} ψ = 0

Has some issues which are solved by the Dirac equation:

it has a second time derivative of the wave function. Therefore, to solve it we must specify not only the initial value of the wave equation, but also the derivative of the wave equation,
As mentioned at Advanced quantum mechanics by Freeman Dyson (1951) and further clarified at: physics.stackexchange.com/questions/340023/cant-the-negative-probabilities-of-klein-gordon-equation-be-avoided, this would lead to negative probabilities.
the modulus of the wave function is not constant and therefore not always one, and therefore cannot be interpreted as a probability density anymore
since we are working with the square of the energy, we have both positive and negative value solutions. This is also a features of the Dirac equation however.

Bibliography:

Video "Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)" at youtu.be/OCuaBmAzqek?t=600
An Introduction to QED and QCD by Jeff Forshaw (1997) 1.2 "Relativistic Wave Equations" and 1.4 "The Klein Gordon Equation" gives some key ideas
2011 PHYS 485 lecture videos by Roger Moore from the University of Alberta at around 7:30
www.youtube.com/watch?v=WqoIW85xwoU&list=PL54DF0652B30D99A4&index=65 "L2. The Klein-Gordon Equation" by doctorphys
sites.ualberta.ca/~gingrich/courses/phys512/node21.html from Advanced quantum mechanics II by Douglas Gingrich (2004)
- sites.ualberta.ca/~gingrich/courses/phys512/node23.html gives Lorentz invariance

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