A phase vocoder is an audio signal processing technique primarily used for time-stretching and pitch-shifting audio signals without significantly altering their quality. It operates based on principles of Fourier analysis and synthesis, and is widely used in electronic music production, sound design, and other audio applications. ### How It Works 1.

An "alternative" formulation of quantum mechanics that does not involve operators.

Is the only atom that has a closed form solution, which allows for very good predictions, and gives awesome intuition about the orbitals in general.

It is arguably the most important solution of the Schrodinger equation.

In particular, it predicts:

The explicit solution can be written in terms of spherical harmonics.

Animal locomotion refers to the various ways in which animals move from one place to another. This movement can involve different body parts and mechanisms, depending on the species and its environment. The modes of locomotion are primarily categorized based on the anatomy and abilities of the animals, as well as the medium in which they move (such as land, water, or air). Common types of animal locomotion include: 1. **Walking and Running**: Most terrestrial animals use legs to walk or run.

That is, two electrons per atomic orbital, each with a different spin.

As shown at Schrödinger equation solution for the helium atom, they do repel each other, and that affects their measurable energy.

However, this energy is still lower than going up to the next orbital. TODO numbers.

Bibliography:

This changes however at higher orbitals, notably as approximately described by the aufbau principle.

The most notable exception is the borrowing of 3d-orbital electrons to 4s as in chromium, leading to a 3d5 4s1 configuration instead of the 3d4 4s2 we would have with the rule. TODO how is that observed observed experimentally?

Higher spin multiplicity means lower energy. I.e.: you want to keep all spins pointin in the same direction.

Logical consequence, often referred to in formal logic as entailment, is a relationship between statements whereby one statement (or set of statements) necessarily follows from another statement (or set of statements). In other words, if a set of premises logically entails a conclusion, then if the premises are true, the conclusion must also be true. In more formal terms, we can express this using symbolic logic.

TODO definition. Appears to be isomers

Example:

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:

There is also a time-energy uncertainty principle, because those two operators are also complementary.

Education has become an expensive bureaucratic exercise, completely dissociated from reality and usefulness.

It completely rejects what the individual wants to achieve, and instead attempts to mass homogenize and test people through endless hours of boredom.

And the only goals it achieves are testing student's resilience to stress, and facilitating the finding of sexual partners. True learning is completely absent.

Teachers only teach because they have to do it to get paid, not for passion. Their only true incentive is co-authoring papers.

We reject this bullshit.

Education is meant to help us, the students, achieve our goals through passionate learning.

And, we, the students, are individuals, with different goals and capabilities.

The way we protest is to publish the knowledge from University for free, on the Internet, so that anyone can access it.

And we do this is a law-abiding way, without copyright infringement, so that no one can legally take it down.

We come to our courses just for the useless roll calls. But we already know all the subject better than the "teacher" on the very first day.

And we are already more famous than the "teacher" online, and through the Internet have already taught more way way more people than they ever will.

The effect of this is to demoralize the entire school system at all levels, until only one conclusion is possible: implosion.

And from the ashes of the old system, we will build a new one, which does only what matters with absolute efficiency: help the individual students achieve their goals.

A system in which the only reason why university exist will be to allow the most knowledgeable students to access million dollar laboratory equipment, and to pay the most prolific content creators so they can continue content creating.

No more useless courses. No more useless tests. Only passion, usefulness and focus.

Contains the full state of the quantum system.

This is in contrast to classical mechanics where e.g. the state of mechanical system is given by two real functions: position and speed.

The wave equation in position representation on the other hand encodes speed in "how fast does the complex phase spin around", and direction in "does it spin clockwise or counterclockwise", as described well at: Video "Visualization of Quantum Physics (Quantum Mechanics) by udiprod (2017)". Then once you understand that, it is more compact to just view those graphs with the phase color coded as in Video "Simulation of the time-dependent Schrodinger equation (JavaScript Animation) by Coding Physics (2019)".

Published by Werner Heisenberg in 1925-07-25 as quantum mechanical re-interpretation of kinematic and mechanical relations by Heisenberg (1925), it offered the first general formulation of quantum mechanics.

It is apparently more closely related to the ladder operator method, which is a more algebraic than the more analytical Schrödinger equation.

It appears that this formulation makes the importance of the Poisson bracket clear, and explains why physicists are so obsessed with talking about position and momentum space. This point of view also apparently makes it clearer that quantum mechanics can be seen as a generalization of classical mechanics through the Hamiltonian.

QED and the men who made it: Dyson, Feynman, Schwinger, and Tomonaga by Silvan Schweber (1994) mentions however that relativistic quantum mechanics broke that analogy, because some 2x2 matrix had a different form, TODO find that again.

Inward Bound by Abraham Pais (1988) chapter 12 "Quantum mechanics, an essay" part (c) "A chronology" has some ultra brief, but worthwhile mentions of matrix mechanics and the commutator.