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Open knowledge by

Ciro Santilli 40 Updated 2025-07-16

Ciro Santilli's raison d'etre, one of his attempts: OurBigBook.com.

The outcome of closed knowledge is reverse engineering.

Photon polarization by

Ciro Santilli 40 Updated 2025-07-16

The knowledge that light is polarized precedes the knowledge of the existence of the photon, see polarization of light for the classical point of view.

The polarization state and how it can be decomposed into different modes can be well visualized with the Poincaré sphere.

One key idea about photon polarization is that it carries angular momentum. Therefore, when an electron changes orbitals in the Schrödinger equation solution for the hydrogen atom, the angular momentum (as well as energy) change is carried out by the polarization of the photon!

Quantum Mechanics 9b - Photon Spin and Schrodinger's Cat II by ViaScience (2013)

Source.

clear animations showing how two circular polarizations can make a vertical polarization
a polarizer can be modelled bra operator.
light polarization experiments are extremely direct evidence of quantum superposition. Individual photons must be on both L and R states at the same time because a V filter passes half of either L or R single photons, but it passes all L + R photons

Optical ring resonator by

Ciro Santilli 40 Updated 2025-07-16

Muon by

Ciro Santilli 40 Updated 2025-07-16

Proton-to-electron mass ratio by

Ciro Santilli 40 Updated 2025-07-16

Kaon by

Ciro Santilli 40 Updated 2025-07-16

One strange quark bound with one up quark or a down quark. 6 combinations exist, 4 if we consider antiparticles the same as particles.

Emission spectrum by

Ciro Santilli 40 Updated 2025-07-16

Adobe Flash by

Ciro Santilli 40 Updated 2025-07-16

Double-slit experiment by

Ciro Santilli 40 Updated 2025-07-16

Amazingly confirms the wave particle duality of quantum mechanics.

The effect is even more remarkable when done with individual particles such individual photons or electrons.

Richard Feynman liked to stress how this experiment can illustrate the core ideas of quantum mechanics. Notably, he night have created the infinitely many slits thought experiment which illustrates the path integral formulation.

Single particle double slit experiment by

Ciro Santilli 40 Updated 2025-07-16

electron experiments: single electron double slit experiment.

This experiment seems to be really hard to do, and so there aren't many super clear demonstration videos with full experimental setup description out there unfortunately.

Wikipedia has a good summary at: en.wikipedia.org/wiki/Double-slit_experiment#Overview

For single-photon non-double-slit experiments see: single photon production and detection experiments. Those are basically a pre-requisite to this.

photon experiments:

aapt.scitation.org/doi/full/10.1119/1.4955173 "Video recording true single-photon double-slit interference" by Aspden and Padgetta (2016). Abstract says using spontaneous parametric down-conversion detection of the second photon to know when to turn the camera on

Non-elementary particle:

2019-10-08: 25,000 Daltons
interactive.quantumnano.at/letsgo/ awesome interactive demo that allows you to control many parameters on a lab. Written in Flash unfortunately, in 2015... what a lack of future proofing!

Single Photon Interference by Veritasium (2013)

Source. Claims to do exactly what we want, but does not describe the setup precisely well enough. Notably, does not justify how he knows that single photons are being produced.

Quantum Hall effect by

Ciro Santilli 40 Updated 2025-07-16

Quantum version of the Hall effect.

As you increase the magnetic field, you can see the Hall resistance increase, but it does so in discrete steps.

Figure 1.
Hall resistance as a function of the applied magnetic field showing the Quantum Hall effect
. Source. As we can see, the blue line of the Hall resistance TODO material, temperature, etc. It is unclear if this is just

Gotta understand this because the name sounds cool. Maybe also because it is used to define the fucking ampere in the 2019 redefinition of the SI base units.

At least the experiment description itself is easy to understand. The hard part is the physical theory behind.

TODO experiment video.

The effect can be separated into two modes:

Integer quantum Hall effect: easier to explain from first principles
Fractional quantum Hall effect: harder to explain from first principles
- Fractional quantum Hall effect for $ν = 1/ m$ : 1998 Nobel Prize in Physics
- Fractional quantum Hall effect for $ν \neq = 1/ m$ : one of the most important unsolved physics problems as of 2023

Integer and fractional quantum Hall effects by Matthew A. Grayson

. Source. Presented 2015. This dude did good.

C standard library by

Ciro Santilli 40 Updated 2025-07-16

Why are complex numbers used in the Schrodinger equation? by

Ciro Santilli 40 Updated 2025-07-16

Ciro's 10 cents: physics.stackexchange.com/questions/32422/qm-without-complex-numbers/557600#557600

Why is there a complex number in the equation? Intuitively and mathematically:

Some ideas:

explicit scalar form of the Maxwell's equations

Necessity of complex numbers in the Schrödinger equation by Barton Zwiebach (2017)

Source.

This useless video doesn't really explain anything, he just says "it's needed because the equation has an

i

in it".

The real explanation is: they are not needed, they just allow us to write the equation in a shorter form, which is always a win: physics.stackexchange.com/questions/32422/qm-without-complex-numbers/557600#557600

Macroscopic quantum phenomena by

Ciro Santilli 40 Updated 2025-07-16

The Principles of Quantum Mechanics by Paul Dirac (1930) by

Ciro Santilli 40 Updated 2025-07-16

Time-independent Schrödinger equation for a free one dimensional particle by

Ciro Santilli 40 Updated 2025-07-16

\frac{\partial ^{2}}{\partial x ^{2}} ψ (x) = - \frac{2 m E}{ℏ ^{2}} ψ (x)

(1)

so the solution is:

ψ (x) = e^{\frac{2 m E i x}{ℏ}}

(2)

We notice that the solution has continuous spectrum, since any value of

E

can provide a solution.

Quantum physics by Jim Branson (2003) by

Ciro Santilli 40 Updated 2025-07-16

quantummechanics.ucsd.edu/ph130a/130_notes/130_notes.html

For the UCSD Physics 130 course.

Last updated: 2013.

Very good! Goes up to the Dirac equation.

There were apparently some lecture videos at: web.archive.org/web/20030604194654/http://physicsstream.ucsd.edu/courses/spring2003/physics130a/ as pointed out by Matthew Heaney ^[ref], .mov files can be found at: web.archive.org/web/*/http://physicsstream.ucsd.edu/courses/spring2003/physics130a/*, but we were yet unable to open them, related:

Fortran by

Ciro Santilli 40 Updated 2025-07-16

Schrödinger equation by

Ciro Santilli 40 Updated 2025-07-16

Existence and uniqueness: mathoverflow.net/questions/212913/existence-and-uniqueness-for-two-dimensional-time-dependent-schr%C3%B6dinger-equation

The partial differential equation of non-relativistic quantum mechanics.

Experiments explained:

via the Schrödinger equation solution for the hydrogen atom it predicts:
- spectral line basic lines, plus Zeeman effect
Schrödinger equation solution for the helium atom: perturbative solutions give good approximations to the energy levels
double-slit experiment: I think we have a closed solution for the max and min probabilities on the measurement wall, and they match experiments

Experiments not explained: those that the Dirac equation explains like:

To get some intuition on the equation on the consequences of the equation, have a look at:

The easiest to understand case of the equation which you must have in mind initially that of the Schrödinger equation for a free one dimensional particle.

Then, with that in mind, the general form of the Schrödinger equation is:

i ℏ \frac{\partial ψ ( x , t )}{\partial t} = \hat{H} [ψ (x, t)]

(1)

Equation 1.

Schrodinger equation

where:

$ℏ$ is the reduced Planck constant
$ψ$ is the wave function
$t$ is the time
$\hat{H}$ is a linear operator called the Hamiltonian. It takes as input a function $ψ$ , and returns another function. This plays a role analogous to the Hamiltonian in classical mechanics: determining it determines what the physical system looks like, and how the system evolves in time, because we can just plug it into the equation and solve it. It basically encodes the total energy and forces of the system.

The

x

argument of

ψ

could be anything, e.g.:

we could have preferred polar coordinates instead of linear ones if the potential were symmetric around a point
we could have more than one particle, e.g. solutions of the Schrodinger equation for two electrons, which would have e.g. $x_{1}$ and $x_{2}$ for different particles. No matter how many particles there are, we have just a single $ψ$ , we just add more arguments to it.
we could have even more generalized coordinates. This is much in the spirit of Hamiltonian mechanics or generalized coordinates

Note however that there is always a single magical

t

time variable. This is needed in particular because there is a time partial derivative in the equation, so there must be a corresponding time variable in the function. This makes the equation explicitly non-relativistic.

The general Schrödinger equation can be broken up into a trivial time-dependent and a time-independent Schrödinger equation by separation of variables. So in practice, all we need to solve is the slightly simpler time-independent Schrödinger equation, and the full equation comes out as a result.

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