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C++ memory model by

Ciro Santilli  40 Updated 2025-07-16

stackoverflow.com/questions/6319146/c11-introduced-a-standardized-memory-model-what-does-it-mean-and-how-is-it-g

How to reference a book in Wikipedia markup? by

Ciro Santilli  40 Updated 2025-07-16

Their reference markup is incredibly overengineered, convoluted, and underdocumented, it is unbelivable!

Use the reference:

This is a fact.{{sfn|Schweber|1994|p=487}}

Define the reference:

===Sources===
{{refbegin|2|indent=yes}}
*{{Cite book|author-link=Silvan S. Schweber |title=QED and the Men Who Made It: Dyson, Feynman, Schwinger, and Tomonaga|last=Schweber|first=Silvan S.|location=Princeton|publisher=University Press|year=1994 |isbn=978-0-691-03327-3 |url=https://archive.org/details/qedmenwhomadeitd0000schw/page/492 |url-access=registration}}
{{refend}}

sfn is magic and matches the the author last name and date from the Cite, it is documented at: en.wikipedia.org/wiki/Template:Sfn

Unforutunately, if there are multiple duplicate Cites inline in the article, it will complain that there are multiple definitions, and you have to first factor out the article by replacing all those existing Cite with sfn, and keeping just one Cite at the bottom. What a pain...

You can also link to a specific page of the book, e.g. if it is a book is on Internet Archive Open Library with:

{{sfn|Murray|1997|p=[https://archive.org/details/supermenstory00murr/page/86 86]}}

For multiple pages should use pp= instead of p=. Does not seem to make much difference on the rendered output besides showing p. vs pp., but so be it:

{{sfn|Murray|1997|pp=[https://archive.org/details/supermenstory00murr/page/86 86-87]}}

Faraday's law of induction by

Ciro Santilli  40 Updated 2025-07-16

Lorentz force by

Ciro Santilli  40 Updated 2025-07-16

force_density = ρ E + J \times B

Equation 1.

Lorentz force

A little suspicious that it bears the name of Lorentz, who is famous for special relativity, isn't it? See: Maxwell's equations require special relativity.

Explicit scalar form of the Maxwell's equations by

Ciro Santilli  40 Updated 2025-07-16

For numerical algorithms and to get a more low level understanding of the equations, we can expand all terms to the simpler and more explicit form:

\frac{\partial E _{x}}{\partial x} + \frac{\partial E _{y}}{\partial x} + \frac{\partial E _{z}}{\partial x} = \frac{ρ}{ε _{0}} \frac{\partial B _{x}}{\partial x} + \frac{\partial B _{y}}{\partial x} + \frac{\partial B _{z}}{\partial x} = 0 \frac{\partial E _{z}}{\partial y} - \frac{\partial E _{y}}{\partial z} = - \frac{\partial B _{x}}{\partial t} \frac{\partial E _{x}}{\partial z} - \frac{\partial E _{z}}{\partial x} = - \frac{\partial B _{y}}{\partial t} \frac{\partial E _{y}}{\partial x} - \frac{\partial E _{x}}{\partial y} = - \frac{\partial B _{z}}{\partial t} \frac{\partial B _{z}}{\partial y} - \frac{\partial B _{y}}{\partial z} = μ_{0} (J_{x} + ε_{0} \frac{\partial E _{x}}{\partial t}) \frac{\partial B _{x}}{\partial z} - \frac{\partial B _{z}}{\partial x} = μ_{0} (J_{y} + ε_{0} \frac{\partial E _{y}}{\partial t}) \frac{\partial B _{y}}{\partial x} - \frac{\partial B _{x}}{\partial y} = μ_{0} (J_{z} + ε_{0} \frac{\partial E _{z}}{\partial t})

Equation 1

Coulomb's law by

Ciro Santilli  40 Updated 2025-07-16

Static case of Maxwell's law for electricity only.

Is implied by Gauss' law if Maxwell's equations: physics.stackexchange.com/questions/44418/are-the-maxwells-equations-enough-to-derive-the-law-of-coulomb

The "static" part is important: if this law were true for moving charges, we would be able to transmit information instantly at infinite distances. This is basically where the idea of field comes in.

Video 1.

Coulomb's Law experiment with torsion balance with a mirror on the balance to amplify rotations by uclaphysics (2010)

Source.

Solutions of Maxwell's equations by

Ciro Santilli  40 Updated 2025-07-16

Video 1.

Understanding Electromagnetic Radiation! by Learn Engineering (2019)

Source. Shows animations of a dipole antenna which illustrates well how radiation is emitted from moving charges and travels at the speed of light.

Maxwell's equations in 2D by

Ciro Santilli  40 Updated 2025-07-16

TODO it would be awesome if we could de-generalize the equations in 2D and do a JavaScript demo of it!

Not sure it is possible though because the curl appears in the equations:

Existence and uniqueness of solutions to Maxwell's equations by

Ciro Santilli  40 Updated 2025-07-16

TODO: I'm surprised that the Wiki page barely talks about it, and there are few Google hits too! A sample one: www.researchgate.net/publication/228928756_On_the_existence_and_uniqueness_of_Maxwell's_equations_in_bounded_domains_with_application_to_magnetotellurics

Electric field by

Ciro Santilli  40 Updated 2025-07-16

Hall effect by

Ciro Santilli  40 Updated 2025-07-16

The voltage changes perpendicular to the current when magnetic field is applied.

Figure 1.
Hall effect experimental diagram
. Source. The Hall effect refers to the produced voltage $V_{H}$ , AKA $V_{y}$ on this setup.

An intuitive video is:

Video 1. Source.

The key formula for it is:

V_{H} = \frac{I _{x} B _{z}}{n t e}

where:

$I_{x}$ : current on x direction, which we can control by changing the voltage $V_{x}$
$B_{z}$ : strength of transversal magnetic field applied
$n$ : charge carrier density, a property of the material used
$t$ : height of the plate
$e$ : electron charge

Applications:

the direction of the effect proves that electric currents in common electrical conductors are made up of negative charged particles
measure magnetic fields, TODO vs other methods

Other more precise non-classical versions:

quantum Hall effect

Bibliography:

hyperphysics.phy-astr.gsu.edu/hbase/magnetic/Hall.html

Electromagnetic four-potential by

Ciro Santilli  40 Updated 2025-07-16

A different and more elegant way to express Maxwell's equations by using the:

instead of the:

Maxwell-Boltzmann statistics by

Ciro Santilli  40 Updated 2025-07-16

Experimental verification of the Maxwell-Boltzmann distribution by

Ciro Santilli  40 Updated 2025-07-16

Most applications of the Maxwell-Boltzmann distribution confirm the theory, but don't give a very direct proof of its curve.

Here we will try to gather some that do.

Every closed subgroup of

G L (n, C)

is a Lie group by

Ciro Santilli  40 Updated 2025-07-16

S page 146.

Lie algebra of a matrix Lie group by

Ciro Santilli  40 Updated 2025-07-16

For this sub-case, we can define the Lie algebra of a Lie group

G

as the set of all matrices

M \in G

such that for all

t \in R

e^{tM} \in G

If we fix a given

M

and vary

t

, we obtain a subgroup of

G

. This type of subgroup is known as a one parameter subgroup.

The immediate question is then if every element of

G

can be reached in a unique way (i.e. is the exponential map a bijection). By looking at the matrix logarithm however we conclude that this is not the case for real matrices, but it is for complex matrices.

Examples:

TODO example it can be seen that the Lie algebra is not closed matrix multiplication, even though the corresponding group is by definition. But it is closed under the Lie bracket operation.

Invertible matrix by

Ciro Santilli  40 Updated 2025-07-16

The set of all invertible matrices forms a group: the general linear group with matrix multiplication. Non-invertible matrices don't form a group due to the lack of inverse.

Magic: The Gathering meta-based deck choice is a bimatrix game by

Ciro Santilli  40 Updated 2025-07-16

Ciro had initially Googled for the "4-card limit thought experiment" but he reached: www.channelfireball.com/articles/what-if-the-4-card-limit-was-abolished-in-modern/ "What if the 4-Card Limit Was Abolished in Modern?" by Frank Karsten (2018) and was much more pleased with the mathematical result. Like-minded people.

That links to www.channelfireball.com/articles/what-if-the-4-card-limit-was-abolished-in-modern/ the related article: "The Mythic Invitational's Duo Standard Format Game Theory Optimized" by Frank Karsten (2019) which explains well how Nash equilibrium is naturally reached: if there is any imbalance, someone can take advantage of it, and then it rebalances. Therefore once you've calculated the equilibrium, your best course of action is to pick a deck at random from a list of possible winners.

Of course, part of the beauty of Magic is that we can never really know the full matrix since deck choice is basically infinite. But at the very least it could give good archetype results.

Schrödinger picture by

Ciro Santilli  40 Updated 2025-07-16

To better understand the discussion below, the best thing to do is to read it in parallel with the simplest possible example: Schrödinger picture example: quantum harmonic oscillator.

The state of a quantum system is a unit vector in a Hilbert space.

"Making a measurement" for an observable means applying a self-adjoint operator to the state, and after a measurement is done:

the state collapses to an eigenvector of the self adjoint operator
the result of the measurement is the eigenvalue of the self adjoint operator
the probability of a given result happening when the spectrum is discrete is proportional to the modulus of the projection on that eigenvector.
For continuous spectra such as that of the position operator in most systems, e.g. Schrödinger equation for a free one dimensional particle, the projection on each individual eigenvalue is zero, i.e. the probability of one absolutely exact position is zero. To get a non-zero result, measurement has to be done on a continuous range of eigenvectors (e.g. for position: "is the particle present between x=0 and x=1?"), and you have to integrate the probability over the projection on a continuous range of eigenvalues.
In such continuous cases, the probability collapses to an uniform distribution on the range after measurement.
The continuous position operator case is well illustrated at: Video "Visualization of Quantum Physics (Quantum Mechanics) by udiprod (2017)"

Those last two rules are also known as the Born rule.

Self adjoint operators are chosen because they have the following key properties:

their eigenvalues form an orthonormal basis
they are diagonalizable

Perhaps the easiest case to understand this for is that of spin, which has only a finite number of eigenvalues. Although it is a shame that fully understanding that requires a relativistic quantum theory such as the Dirac equation.

The next steps are to look at simple 1D bound states such as particle in a box and quantum harmonic oscillator.

This naturally generalizes to Schrödinger equation solution for the hydrogen atom.

The solution to the Schrödinger equation for a free one dimensional particle is a bit harder since the possible energies do not make up a countable set.

This formulation was apparently called more precisely Dirac-von Neumann axioms, but it because so dominant we just call it "the" formulation.

Quantum Field Theory lecture notes by David Tong (2007) mentions that:

if you were to write the wavefunction in quantum field theory, it would be a functional, that is a function of every possible configuration of the field $ϕ$ .