Ciro Santilli @cirosantilli 35

The discovery of the photon was one of the major initiators of quantum mechanics.

Light was very well known to be a wave through diffraction experiments. So how could it also be a particle???

This was a key development for people to eventually notice that the electron is also a wave.

This process "started" in 1900 with Planck's law which was based on discrete energy packets being exchanged as exposed at On the Theory of the Energy Distribution Law of the Normal Spectrum by Max Planck (1900).

This ideas was reinforced by Einstein's explanation of the photoelectric effect in 1905 in terms of photon.

In the next big development was the Bohr model in 1913, which supposed non-classical physics new quantization rules for the electron which explained the hydrogen emission spectrum. The quantization rule used made use of the Planck constant, and so served an initial link between the emerging quantized nature of light, and that of the electron.

The final phase started in 1923, when Louis de Broglie proposed that in analogy to photons, electrons might also be waves, a statement made more precise through the de Broglie relations.

This event opened the floodgates, and soon matrix mechanics was published in quantum mechanical re-interpretation of kinematic and mechanical relations by Heisenberg (1925), as the first coherent formulation of quantum mechanics.

It was followed by the Schrödinger equation in 1926, which proposed an equivalent partial differential equation formulation to matrix mechanics, a mathematical formulation that was more familiar to physicists than the matrix ideas of Heisenberg.

Inward Bound by Abraham Pais (1988) summarizes his views of the main developments of the subjectit:

Bibliography:

For some reason, Ciro Santilli is mildly obsessed with understanding and visualizing the real projective plane.

To see why this is called a plane, move he center of the sphere to $z=1$, and project each line passing on the center of the sphere on the x-y plane. This works for all points of the sphere, except those at the equator $z=1$. Those are the points at infinity. Note that there is one such point at infinity for each direction in the x-y plane.

On Baidu Baike: baike.baidu.com/item/扇画/8486689

This is perhaps slightly worse than the Tinker Tailor Soldier Spy, but still amazing.

Some difficult points:

1TB.

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www.seagate.com/www-content/datasheets/pdfs/mobile-hddDS1861-2-1603-en_US.pdf | web.archive.org/web/20181225095438/https://www.seagate.com/www-content/datasheets/pdfs/mobile-hddDS1861-2-1603-en_US.pdf

sudo hdparm -Tt /dev/sda3 on Ubuntu 20.04:

Nominal maximum speed: 140 MB/s

Examples: square, octahedron.

Take $(0,0,0,\dots ,0)$ and flip one of 0's to $\pm 1$. Therefore has $2\times D$ vertices.

Each edge E is linked to every other edge, except it's opposite -E.

Proportionality factor in the Planck-Einstein relation between light energy and frequency.

And analogously for matter, appears in the de Broglie relations relating momentum and frequency. Also appears in the Schrödinger equation, basically as a consequence/cause of the de Broglie relations most likely.

Intuitively, the Planck constant determines at what length scale do quantum effects start to show up for a given energy scale. It is because the Plank constant is very small that we don't perceive quantum effects on everyday energy/length/time scales. On the ${\lim }_{h\to 0}$, quantum mechanics disappears entirely.

A very direct way of thinking about it is to think about what would happen in a double-slit experiment. TODO think more clearly what happens there.

Defined exactly in the 2019 redefinition of the SI base units to:

A superstar security researcher with some major exploits from in the 2000's.

Analogous problem to the secondary structure of proteins. Likely a bit simpler due to the strong tendency for complementary pairs to bind.

The $U(1)$ unitary group is one very over-generalized way of looking at it :-)

Welcome to the wonderful world of Cirism!

Followers of Cirism call themselves Cirists, and their primary goal in life is to obtain Cirocoins.

Cirism is totally not a cult, has been officially verified to be compatible with all major world religions.

Enlightened Cirists donate money to the cause at: Section "Sponsor Ciro Santilli's work on OurBigBook.com". It is totally optional of course, your soul will just be eternally damned if you don't.

Ciro Santilli once proclaimed:

The main interest of this theorem is in classifying the indefinite orthogonal groups, which in turn is fundamental because the Lorentz group is an indefinite orthogonal groups, see: all indefinite orthogonal groups of matrices of equal metric signature are isomorphic.

It also tells us that a change of basis does not the alter the metric signature of a bilinear form, see matrix congruence can be seen as the change of basis of a bilinear form.

The theorem states that the number of 0, 1 and -1 in the metric signature is the same for two symmetric matrices that are congruent matrices.

For example, consider:

The eigenvalues of $A$ are $1$ and $4$, and the associated eigenvectors are:symPy code:and from the eigendecomposition of a real symmetric matrix we know that:

Now, instead of $P$, we could use $PE$, where $E$ is an arbitrary diagonal matrix of type:With this, would reach a new matrix $B$:Therefore, with this congruence, we are able to multiply the eigenvalues of $A$ by any positive number $e_1^2$ and $e_2^2$. Since we are multiplying by two arbitrary positive numbers, we cannot change the signs of the original eigenvalues, and so the metric signature is maintained, but respecting that any value can be reached.

Note that the matrix congruence relation looks a bit like the eigendecomposition of a matrix:but note that $D$ does not have to contain eigenvalues, unlike the eigendecomposition of a matrix. This is because here $S$ is not fixed to having eigenvectors in its columns.

But because the matrix is symmetric however, we could always choose $S$ to actually diagonalize as mentioned at eigendecomposition of a real symmetric matrix. Therefore, the metric signature can be seen directly from eigenvalues.

Also, because $D$ is a diagonal matrix, and thus symmetric, it must be that:

What this does represent, is a general change of basis that maintains the matrix a symmetric matrix.

Related:

As always, the best way to get some intuition about an equation is to solve it for some simple cases, so let's give that a try with different fixed potentials.

The wave equation contains the entire state of a particle.

From mathematical formulation of quantum mechanics remember that the wave equation is a vector in Hilbert space.

And a single vector can be represented in many different ways in different basis, and two of those ways happen to be the position and the momentum representations.

More importantly, position and momentum are first and foremost operators associated with observables: the position operator and the momentum operator. And both of their eigenvalue sets form a basis of the Hilbert space according to the spectral theorem.

When you represent a wave equation as a function, you have to say what the variable of the function means. And depending on weather you say "it means position" or "it means momentum", the position and momentum operators will be written differently.

This is well shown at: Video "Visualization of Quantum Physics (Quantum Mechanics) by udiprod (2017)".

Furthermore, the position and momentum representations are equivalent: one is the Fourier transform of the other: position and momentum space. Remember that notably we can always take the Fourier transform of a function in $L^2$ due to Carleson's theorem.

Then the uncertainty principle follows immediately from a general property of the Fourier transform: en.wikipedia.org/w/index.php?title=Fourier_transform&oldid=961707157#Uncertainty_principle

In precise terms, the uncertainty principle talks about the standard deviation of two measures.

We can visualize the uncertainty principle more intuitively by thinking of a wave function that is a real flat top bump function with a flat top in 1D. We can then change the width of the support, but when we do that, the top goes higher to keep probability equal to 1. The momentum is 0 everywhere, except in the edges of the support. Then:

Bibliography:

What happens when the underdogs get together and try to factor out their efforts to beat some evil dominant power, sometimes victoriously.

Or when startups use the cheapest stuff available and randomly become the next big thing, and decide to keep maintaining the open stuff to get features for free from other companies, or because they are forced by the Holy GPL.

Open source frees employees. When you change jobs, a large part of the specific knowledge you acquired about closed source a project with your blood and tears goes to the trash. When companies get bought, projects get shut down, and closed source code goes to the trash. What sane non desperate person would sell their life energy into such closed source projects that could die at any moment? Working on open source is the single most important non money perk a company can have to attract the best employees.

Open source is worth more than the mere pragmatic financial value of not having to pay for software or the ability to freely add new features.

Its greatest value is perhaps the fact that it allows people study it, to appreciate the beauty of the code, and feel empowered by being able to add the features that they want.

That is why Ciro Santilli thought:

But quoting Ciro's colleague S.:

And "can reverse engineer the undocumented GPU hardware APIs", Ciro would add.

While software is the most developed open source technology available in the 2010's, due to the "zero cost" of copying it over the Internet, Ciro also believes that the world would benefit enormously from open source knowledge in all areas on science and engineering, for the same reasons as open source.