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.
"Making a measurement" for an observable means applying a self-adjoint operator to the state, and after a measurement is done:Those last two rules are also known as the Born rule.
- 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)"
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 .
An "alternative" formulation of quantum mechanics that does not involve operators.
Implementations:
- Hall effect based, i.e. a Hall effect sensor
- SQUID device
is by far the most important of because it is quantum mechanics states live, because the total probability of being in any state has to be 1!
has some crucially important properties that other don't (TODO confirm and make those more precise):
- it is the only that is Hilbert space because it is the only one where an inner product compatible with the metric can be defined:
- Fourier basis is complete for , which is great for solving differential equation
Early electron diffraction experiment from 1927 that drastically confirmed the matter wave hypothesis.
Niels Bohr for the Bohr model.
The questions are: who is this Mark, and why does he have to go down?
Cryptocurrency with focus on anonymity. Was almost certainly the leading privacy coin since its inception until as of writing in the 2020s.
Ciro Santilli has received and held considerable quantities of Monero, notably 1000 Monero donation. so bias alert.
As mentioned at Section "Are cryptocurrencies useful?", Ciro Santilli believes that anonymity is the most valuable feature that really matters on crypto coins, and therefore if he were to invest in crypto, he would invest in Monero or some other privacy coin.
localmonero.co/knowledge/monero-stealth-addresses?language=en gives an overview of the privacy mechanisms:
- ring signatures, which hide the true output (sender)localmonero.co/knowledge/ring-signatures Gives an overview. Mentions that it is prone to heuristic attacks.Uses a system of decoys, that adds 10 fake possible previous outputs as inputs, in addition to the actual input.So the network only knows/verifies that one of those 11 previous outputs was used, but it does not know which one.TODO so how do you know which previous outputs were spent or not?
- RingCT which hides the amounts.
- stealth addresses, which hides who you send toThis forces receivers to scan try and unlock every single transaction in the chain to see if it is theirs or not.The sender therefore can know when the money is spent, but once again, not to whom it is being sent.
Based on en.wikipedia.org/wiki/CryptoNote and like Satoshi Nakamoto created by under the pseudonym "Nicolas van Saberhagen" www.reddit.com/r/Monero/comments/7v2obe/offering_a_bounty_for_a_video_of_the_speech_by/
Coinbase has actually stayed away from trading it even as of 2019 when Monero was the third largest market capitalization crypto because of fear of regulatory slashback: decrypt.co/36731/heres-why-coinbase-still-hasnt-listed-monero. Although it must be said, the value of privacy crypto is greatly reduced when everyone is trading it on exchanges, which require a passport upload to work.
Surprisingly, it can also become a superfluid even though each atom is a fermion! This is because of Cooper pair formation, just like in superconductors, but the transition happens at lower temperatures than superfluid helium-4, which is a boson.
aps.org/publications/apsnews/202110/history.cfm: October 1972: Publication of Discovery of Superfluid Helium-3 contains comments on the seminal paper and a graph which we must steal.
A solution to Laplace's equation.
There are unlisted articles, also show them or only show them.