Ciro Santilli @cirosantilli 37

Notation used in quantum mechanics.

Ket is just a vector. Though generally in the context of quantum mechanics, this is an infinite dimensional vector in a Hilbert space like $L^2$.

Bra is just the dual vector corresponding to a ket, or in other words projection linear operator, i.e. a linear function which can act on a given vector and returns a single complex number. Also known as... dot product.

For example:is basically a fancy way of saying:that is: we are taking the projection of $y$ along the $x$ direction. Note that in the ordinary dot product notation however, we don't differentiate as clearly what is a vector and what is an operator, while the bra-ket notation makes it clear.

The projection operator is completely specified by the vector that we are projecting it on. This is why the bracket notation makes sense.

It also has the merit of clearly differentiating vectors from operators. E.g. it is not very clear in $x\cdot y$ that $x$ is an operator and $y$ is a vector, except due to the relative position to the dot. This is especially bad when we start manipulating operators by themselves without vectors.

This notation is widely used in quantum mechanics because calculating the probability of getting a certain outcome for an experiment is calculated by taking the projection of a state on one an eigenvalue basis vector as explained at: Section "Mathematical formulation of quantum mechanics".

Making the projection operator "look like a thing" (the bra) is nice because we can add and multiply them much like we can for vectors (they also form a vector space), e.g.:just means taking the projection along the $x+y$ direction.

Ciro Santilli thinks that this notation is a bit over-engineered. Notably the bra's are just vectors, which we should just write as usual with $\overrightarrow{v}$... the bra thing makes it look scarier than it needs to be. And then we should just find a different notation for the projection part.

Maybe Dirac chose it because of the appeal of the women's piece of clothing: bra, in an irresistible call from British humour.

But in any case, alas, we are now stuck with it.

One of the defining properties of algebraic structure with two operations such as ring and field:This property shows how the two operations interact.

See: math.stackexchange.com/questions/579453/real-world-application-of-fourier-series/3729366#3729366

Ciro Santilli has sometimes wasted time with low impact projects such as those listed at Ciro Santilli's minor projects instead of doing higher impact projects such as those mentioned at: Section "The most important projects Ciro Santilli wants to do".

But maybe "Everything you did brought you where you are now." applies, maybe it is during the "low impact activities" that one gets the inspiration and experience required for the "high impact ones".

Ciro Santilli is very happy to meet people with related interests, he really loves his like-minded online friends. Even if you don't have something a specific goal in mind for the contact, please just say hi.

To contact Ciro publicly about any general subject that is not covered in a more specific GitHub repository, including saying hi or suggestions about his website either:

For comments about China, first read:and then create a GitHub issue at: github.com/cirosantilli/china-dictatorship/issues/new

Publicly viewable contact is preferred if possible to more effectively share Ciro's wisdom with the world.

But if you feel more comfortable with private contact, no problem, either:

For other less good methods that will also work, use direct messages of the following profiles from under Section "Accounts controlled by Ciro Santilli":

If you are a privacy freak or are going to tell Ciro state secrets Ciro has this GNU Privacy Guard public key: pubkey.gpg.

Disqus comments were removed from his website in 2019-05-04, a manual dump is available here, removal rationale at: why Ciro Santilli removed Disqus comments from his website in 2019-05-04.

Some authors use the convention of:

Great reports to how 2022-Ciro Santilli views how OurBigBook.com could go. Ciro can't help to feel that he is a mixture of both of the vision, tech and people guy. Not as extreme as any of them, but more like a well rounded (and less good individually) version of each. High flying bird vs gophers.

Reviews:

They are sorted in order of "most likely to get done first".

Top one: OurBigBook.com.

A 4D gradient with some small special relativity specifics added in (the light of speed and sign change for the time).

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:

This book series appears to be the one: global.oup.com/academic/content/series/h/history-of-the-university-of-oxford-huo/. A mere 250 pounds+ each.

There are a few related concepts that are called infinity in mathematics:

Some notable examples:

Subcase of antisymmetric multilinear map: