Dirac equation

Adds special relativity to the Schrödinger equation, and the following conclusions come basically as a direct consequence of this!

Experiments explained:

Experiments not explained: those that quantum electrodynamics explains like:See also: Dirac equation vs quantum electrodynamics.

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.

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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:

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.

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First postulated by Einstein in 1917 on his paper Zur Quantentheorie der Strahlung" ("On the Quantum Theory of Radiation") as a more elegant way to rederive Planck's law as part of the Einstein coefficients framework.

At that time there was no other physical evidence supporting the existence of the concept except that it looked more elegant.

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Shown at: Video "Quantum Mechanics 12b - Dirac Equation II by ViaScience (2015)".

Predicted by the Dirac equation.

Can be easily seen from the solution of Equation "Expanded Dirac equation in Planck units" when the particle is at rest as shown at Video "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.

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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:

Has some issues which are solved by the Dirac equation:

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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:

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:

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.

Related: Dirac equation vs quantum electrodynamics.

Predicts fine structure.

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

Originally done with (neutral) silver atoms 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

I like relativistic quantum mechanics.

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:

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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:

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!

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The name actually comes from "any". Amazing.

Can only exist in 2D surfaces, not 3D, where fermions and bosons are the only options.

All known anyons are quasiparticles.

On particle exchange:so it is a generalization of bosons and fermions which have $\theta =0$ and $\theta =\pi$ respectively.

Key physical experiment: fractional quantum Hall effect.

Exotic and hard to find experimentally.

Video "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.

where:

Remember that $\psi$ 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 $\psi$ 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.