= Dirac equation
{c}
{wiki}
Adds <special relativity> to the <Schrödinger equation>, and the following conclusions come basically as a direct consequence of this!
Experiments explained:
* <spontaneous emission> coefficients.
* <fine structure>, notably for example <Dirac equation solution for the hydrogen atom>
* <antimatter>
* <particle creation and annihilation>
Experiments not explained: those that <quantum electrodynamics> explains like:
* <Lamb shift>
* TODO: quantization of the electromagnetic field as <photons>?
See also: <Dirac equation vs quantum electrodynamics>.
The Dirac equation is a set of 4 <partial differential equations> on 4 <complex number>[complex valued] wave functions. The full explicit form in <Planck units> is shown e.g. in <video Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)> at https://youtu.be/OCuaBmAzqek?t=1010[]:
$$
i \partial_t \begin{bmatrix} \psi_1 \\ \psi_2 \\ \psi_3 \\ \psi_4 \end{bmatrix} =
- i \partial_x \begin{bmatrix} \psi_4 \\ \psi_3 \\ \psi_2 \\ \psi_1 \end{bmatrix}
+ \partial_y \begin{bmatrix}-\psi_4 \\ \psi_3 \\ -\psi_2 \\ \psi_1 \end{bmatrix}
- i \partial_z \begin{bmatrix} \psi_3 \\ -\psi_4 \\ \psi_1 \\ -\psi_2 \end{bmatrix}
+ m \begin{bmatrix} \psi_1 \\ \psi_2 \\ -\psi_3 \\ -\psi_4 \end{bmatrix}
$$
{title=Expanded <Dirac equation> in <Planck units>}
Then as done at https://physics.stackexchange.com/questions/32422/qm-without-complex-numbers/557600\#557600 from <why are complex numbers used in the Schrodinger equation?>, we could further split those equations up into a system of 8 equations on 8 <real number>[real-valued] functions.
\Video[http://youtube.com/watch?v=OCuaBmAzqek]
{title=Quantum Mechanics 12a - Dirac Equation I by <ViaScience> (2015)}
\Video[https://www.youtube.com/watch?v=ajMaPc022VM]
{title=PHYS 485 Lecture 14: The Dirac Equation by <2011 PHYS 485 lecture videos by Roger Moore from the University of Alberta>[Roger Moore] (2016)}