= Klein-Gordon equation
{c}
{wiki=Klein–Gordon_equation}
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 <klein-Gordon equation in Einstein notation>{full} with <Einstein notation> and <Planck units>:
$$
\partial_i \partial^i \psi - m^2 \psi = 0
$$
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: https://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 https://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
* https://www.youtube.com/watch?v=WqoIW85xwoU&list=PL54DF0652B30D99A4&index=65 "L2. The Klein-Gordon Equation" by doctorphys
* https://sites.ualberta.ca/~gingrich/courses/phys512/node21.html from <Advanced quantum mechanics II by Douglas Gingrich (2004)>
* https://sites.ualberta.ca/~gingrich/courses/phys512/node23.html gives <Lorentz invariance>