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:See also: Dirac equation vs quantum electrodynamics.
- Lamb shift
- TODO: quantization of the electromagnetic field as photons?
The Dirac equation is a set of 4 partial differential equations on 4 complex valued wave functions. The full explicit form in Planck units is shown e.g. in Video 1. "Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)" at youtu.be/OCuaBmAzqek?t=1010:Then as done at 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-valued functions.
Equation 1.
Expanded Dirac equation in Planck units
. Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)
Source. PHYS 485 Lecture 14: The Dirac Equation by Roger Moore (2016)
Source. Second quantization also appears to be useful not only for relativistic quantum mechanics, but also for condensed matter physics. The reason is that the basis idea is to use the number occupation basis. This basis is:
- convenient for quantum field theory because of particle creation and annihilation changes the number of particles all the time
- convenient for condensed matter physics because there you have a gazillion particles occupying entire energy bands
Bibliography:
- www.youtube.com/watch?v=MVqOfEYzwFY "How to Visualize Quantum Field Theory" by ZAP Physics (2020). Has 1D simulations on a circle. Starts towards the right direction, but is a bit lacking unfortunately, could go deeper.