The first really good quantum mechanics theory made compatible with special relativity was the Dirac equation.

And then came quantum electrodynamics to improve it: Dirac equation vs quantum electrodynamics.

TODO: does it use full blown QED, or just something intermediate?

www.youtube.com/watch?v=NtnsHtYYKf0 "Mercury and Relativity - Periodic Table of Videos" by Periodic Videos (2013). Doesn't give the key juicy details/intuition. Also mentioned on Wikipedia: en.wikipedia.org/wiki/Relativistic_quantum_chemistry#Mercury

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.

$i∂_{t}⎣⎢⎢⎢⎡ ψ_{1}ψ_{2}ψ_{3}ψ_{4} ⎦⎥⎥⎥⎤ =−i∂_{x}⎣⎢⎢⎢⎡ ψ_{4}ψ_{3}ψ_{2}ψ_{1} ⎦⎥⎥⎥⎤ +∂_{y}⎣⎢⎢⎢⎡ −ψ_{4}ψ_{3}−ψ_{2}ψ_{1} ⎦⎥⎥⎥⎤ −i∂_{z}⎣⎢⎢⎢⎡ ψ_{3}−ψ_{4}ψ_{1}−ψ_{2} ⎦⎥⎥⎥⎤ +m⎣⎢⎢⎢⎡ ψ_{1}ψ_{2}−ψ_{3}−ψ_{4} ⎦⎥⎥⎥⎤ $

Do electrons spontaneously jump from high orbitals to lower ones emitting photons?

Explaining this was was one of the key initial achievements of the Dirac equation.

Yes, but this is not predicted by the Schrödinger equation, you need to go to the Dirac equation.

A critical application of this phenomena is laser.

See also:

- physics.stackexchange.com/questions/233330/why-do-electrons-jump-between-orbitals
- physics.stackexchange.com/questions/117417/quantum-mechanics-scattering-theory/522220#522220
- physics.stackexchange.com/questions/430268/stimulated-emission-how-can-giving-energy-to-electrons-make-them-decay-to-a-low/430288

TODO understand better, mentioned e.g. at Subtle is the Lord by Abraham Pais (1982) page 20, and is something that Einstein worked on.

Photon hits excited electron, makes that electron go down, and generates a new identical photon in the process, with the exact same:This is the basis of lasers.

- frequency
- polarization
- direction

Bibliography:

Predicted by the Dirac equation.

Can be easily seen from the solution of Equation 1. "Expanded Dirac equation in Planck units" when the particle is at rest as shown at Video 4. "Quantum Mechanics 12b - Dirac Equation II by ViaScience (2015)".

Predicted by the Dirac equation.

We've likely known since forever that photons are created: just turn on a light and see gazillion of them come out!

Photon creation is easy because photons are massless, so there is not minimum energy to create them.

The creation of other particles is much rarer however, and took longer to be discovered, one notable milestone being the discovery of the positron.

In the case of the electron, we need to start with at least enough energy for the mass of the electron positron pair. This requires a photon with wavelength in the picometer range, which is not common in the thermal radiation of daily life.

Can produce two entangled particles.

Described for example in lecture 1.

TODO, including why the Schrodinger equation is not.

The Dirac equation can be derived basically "directly" from the Representation theory of the Lorentz group for the spin half representation, this is shown for example at Physics from Symmetry by Jakob Schwichtenberg (2015) 6.3 "Dirac Equation".

The Diract equation is the spacetime symmetry part of the quantum electrodynamics Lagrangian, i.e. is describes how spin half particles behave without interactions. The full quantum electrodynamics Lagrangian can then be reached by adding the $U(1)$ internal symmetry.

As mentioned at spin comes naturally when adding relativity to quantum mechanics, this same method allows us to analogously derive the equations for other spin numbers.

Bibliography:

- Video 1. "Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)" at youtu.be/OCuaBmAzqek?t=743
- www.youtube.com/watch?v=zM-Lc16nyho&list=PL54DF0652B30D99A4&index=66 "L3. The Dirac Equation" by doctorphys
- Video 5. "Dirac equation for the electron and hydrogen Hamiltonian by Barton Zwiebach (2019)"

Bibliography:

- www.youtube.com/watch?v=Fu1BGGeyqHQ&list=PL54DF0652B30D99A4&index=63 "K6. The Pauli Equation" by doctorphys

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:

$∂_{i}∂_{i}ψ−m_{2}ψ=0$

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 1. "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)

The Klein-Gordon equation directly uses a more naive relativistic energy guess of $p_{2}+m_{2}$ squared.

But since this is quantum mechanics, we feel like making $p$ into the "momentum operator", just like in the Schrödinger equation.

But we don't really know how to apply the momentum operator twice, because it is a gradient, so the first application goes from a scalar field to the vector field, and the second one...

So we just cheat and try to use the laplace operator instead because there's some squares on it:

$H=∇_{2}+m_{2}$

But then, we have to avoid taking the square root to reach a first derivative in time, because we don't know how to take the square root of that operator expression.

So the Klein-Gordon equation just takes the approach of using this squared Hamiltonian instead.

Since it is a Hamiltonian, and comparing it to the Schrödinger equation which looks like:
taking the Hamiltonian twice leads to:

$Hψ=i∂t∂ψ $

$H_{2}ψ=−∂_{2}t∂_{2}ψ $

We can contrast this with the Dirac equation, which instead attempts to explicitly construct an operator which squared coincides with the relativistic formula: derivation of the Dirac equation.

Predicts fine structure.

Spin is one of the defining properties of elementary particles, i.e. number that describes how an elementary particle behaves, much like electric charge and mass.

Possible values are half integer numbers: 0, 1/2, 1, 3/2, and so on.

The approach shown in this section: Section "Spin comes naturally when adding relativity to quantum mechanics" shows what the spin number actually means in general. As shown there, the spin number it is a direct consequence of having the laws of nature be Lorentz invariant. Different spin numbers are just different ways in which this can be achieved as per different Representation of the Lorentz group.

Video 7. "Quantum Mechanics 9a - Photon Spin and Schrodinger's Cat I by ViaScience (2013)" explains nicely how:

- incorporated into the Dirac equation as a natural consequence of special relativity corrections, but not naturally present in the Schrödinger equation, see also: the Dirac equation predicts spin
- photon spin can be either linear or circular
- the linear one can be made from a superposition of circular ones
- straight antennas produce linearly polarized photos, and Helical antennas circularly polarized ones
- a jump between 2s and 2p in an atom changes angular momentum. Therefore, the photon must carry angular momentum as well as energy.
- cannot be classically explained, because even for a very large estimate of the electron size, its surface would have to spin faster than light to achieve that magnetic momentum with the known electron charge
- as shown at Video 4. "Quantum Mechanics 12b - Dirac Equation II by ViaScience (2015)", observers in different frames of reference see different spin states

- Stern-Gerlach experiment
- fine structure split in energy levels
- anomalous Zeeman effect
- of a more statistical nature, but therefore also macroscopic and more dramatically observable:
- ferromagnetism
- Bose-Einstein statistics vs Fermi-Dirac statistics. A notable example is the difference in superfluid transition temperature between superfluid helium-3 and superfluid helium-4.

Originally done with silver in 1921, but even clearer theoretically was the hydrogen reproduction in 1927 by T.E. Phipps and J.B. Taylor.

The hydrogen experiment was apparently harder to do and the result is less visible, TODO why: physics.stackexchange.com/questions/33021/why-silver-atoms-were-used-in-stern-gerlach-experiment

Needs an inhomogenous magnetic field to move the atoms up or down: magnetic dipole in an inhomogenous magnetic field. TODO how it is generated?

Basic component in spintronics, used in both giant magnetoresistance

Best mathematical explanation: Section "Spin comes naturally when adding relativity to quantum mechanics".

Physics from Symmetry by Jakob Schwichtenberg (2015) chapter 3.9 "Elementary particles" has an amazing summary of the preceding chapters the spin value has a relation to the representations of the Lorentz group, which encodes the spacetime symmetry that each particle observes. These symmetries can be characterized by small integer numbers:

As usual, we don't know why there aren't elementary particles with other spins, as we could construct them.

Leads to the Klein-Gordon equation.

Leads to the Dirac equation.

Leads to the Proca equation.

Theorized for the graviton.

More interestingly, how is that implied by the Stern-Gerlach experiment?

physics.stackexchange.com/questions/266359/when-we-say-electron-spin-is-1-2-what-exactly-does-it-mean-1-2-of-what/266371#266371 suggests that half could either mean:

- at limit of large
`l`

for the Schrödinger equation solution for the hydrogen atom the difference between each angular momentum is twice that of the eletron's spin. Not very satisfactory. - it comes directly out of the Dirac equation. This is satisfactory. :-)

Initially a phenomenological guess to explain the periodic table. Later it was apparently proven properly with the spin-statistics theorem, physics.stackexchange.com/questions/360140/theoretical-proof-of-paulis-exclusion-principle.

And it was understood more and more that basically this is what prevents solids from collapsing into a single nucleus, not electrical repulsion: electron degeneracy pressure!

Bibliography:

- www.youtube.com/watch?v=EK_6OzZAh5k How Electron Spin Makes Matter Possible by PBS Space Time (2021)

Video 13. "The Biggest Ideas in the Universe | 17. Matter by Sean Carroll (2020)" at youtu.be/dQWn9NzvX4s?t=3707 says that no one has ever been able to come up with an intuitive reason for the proof.

$L=ψˉ (iℏc∂/−mc_{2})ψ$

- $∂$: Feynman slash notation
- $ψˉ $: Dirac adjoint

Remember that $ψ$ is a 4-vetor, gamma matrices are 4x4 matrices, so the whole thing comes down to a dot product of two 4-vectors, with a modified $ψ$ by matrix multiplication/derivatives, and the result is a scalar, as expected for a Lagrangian.

Like any other Lagrangian, you can then recover the Dirac equation, which is the corresponding equations of motion, by applying the Euler-Lagrange equation to the Lagrangian.