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Quantum mechanics is quite a broad term. Perhaps it is best to start approaching it from the division into:
Key experiments that could not work without quantum mechanics: Section "Quantum mechanics experiment".
Mathematics: there are a few models of increasing precision which could all be called "quantum mechanics":
Ciro Santilli feels that the largest technological revolutions since the 1950's have been quantum related, and will continue to be for a while, from deeper understanding of chemistry and materials to quantum computing, understanding and controlling quantum systems is where the most interesting frontier of technology lies.
Atoms exist and last for a long time, while in classical electromagnetic theory punctual orbiting electrons should emit radiation quickly and fall into the nucleus: physics.stackexchange.com/questions/20003/why-dont-electrons-crash-into-the-nuclei-they-orbit
In other sections:
Bibliography:
A single line in the emission spectrum.
So precise, so discrete, which makes no sense in classical mechanics!
Has been the leading motivation of the development of quantum mechanics, all the way from the:
NIST database for spectral line: physics.nist.gov/PhysRefData/ASD/lines_form.html
Let's do a sanity check.
Searching for "H" for hydrogen leads to: physics.nist.gov/cgi-bin/ASD/lines1.pl?spectra=H&limits_type=0&low_w=&upp_w=&unit=1&submit=Retrieve+Data&de=0&format=0&line_out=0&en_unit=0&output=0&bibrefs=1&page_size=15&show_obs_wl=1&show_calc_wl=1&unc_out=1&order_out=0&max_low_enrg=&show_av=2&max_upp_enrg=&tsb_value=0&min_str=&A_out=0&intens_out=on&max_str=&allowed_out=1&forbid_out=1&min_accur=&min_intens=&conf_out=on&term_out=on&enrg_out=on&J_out=on
From there we can see for example the 1 to 2 lines:
  • 1s to 2p: 121.5673644608 nm
  • 1s to 2: 121.56701 nm TODO what does that mean?
  • 1s to 2s: 121.5673123130200 TODO what does that mean?
We see that the table is sorted from lower from level first and then by upper level second.
So it is good to see that we are in the same 121nm ballpark as mentioned at hydrogen spectral line.
TODO why I can't see 2s to 2p transitions there to get the fine structure?
Bibliography:
phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_9HE_-_Modern_Physics/06%3A_Emission_and_Absorption_of_Photons/6.2%3A_Selection_Rules_and_Transition_Times has some very good mentions:
So it appears that if a hydrogen atom emits a photon, it not only has to transition between two states whose energy difference matches the energy of the photon, but it is restricted in other ways as well, if its mode of radiation is to be dipole. For example, a hydrogen atom in its 3p state must drop to either the n=1 or n=2 energy level, to make the energy available to the photon. The n=2 energy level is 4-fold degenerate, and including the single n=1 state, the atom has five different states to which it can transition. But three of the states in the n=2 energy level have l=1 (the 2p states), so transitioning to these states does not involve a change in the angular momentum quantum number, and the dipole mode is not available.
So what's the big deal? Why doesn't the hydrogen atom just use a quadrupole or higher-order mode for this transition? It can, but the characteristic time for the dipole mode is so much shorter than that for the higher-order modes, that by the time the atom gets around to transitioning through a higher-order mode, it has usually already done so via dipole. All of this is statistical, of course, meaning that in a large collection of hydrogen atoms, many different modes of transitions will occur, but the vast majority of these will be dipole.
It turns out that examining details of these restrictions introduces a couple more. These come about from the conservation of angular momentum. It turns out that photons have an intrinsic angular momentum (spin) magnitude of , which means whenever a photon (emitted or absorbed) causes a transition in a hydrogen atom, the value of l must change (up or down) by exactly 1. This in turn restricts the changes that can occur to the magnetic quantum number: can change by no more than 1 (it can stay the same). We have dubbed these transition restrictions selection rules, which we summarize as:
A fundamental component of three-level lasers.
As mentioned at youtu.be/_JOchLyNO_w?t=581 from Video "How Lasers Work by Scientized (2017)", they stay in that state for a long time compared to non spontaneous emission of metastable states!
phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_9HE_-_Modern_Physics/06%3A_Emission_and_Absorption_of_Photons/6.3%3A_Lasers mentions that they are kept in that excited state due to selection rules.
One reasonable and memorable approximation excluding any fine structure is:
Equation 2. Hydrogen spectral series mnemonic
see for example example: hydrogen 1-2 spectral line.
Equation 2. "Hydrogen spectral series mnemonic" gives for example from principal quantum number 1 to 2 a difference:
which with Planck-Einstein relation gives about 121.6 nm ( Hz), which is a reasonable match with the value of 121.567... from the NIST Atomic Spectra Database.
Kind of a synonym for hydrogen emission spectrum not very clear if fine structure is considered by this term or not.
A line set for hydrogen spectral line.
Formula discovered in 1885, was it the first set to have an empirical formula?
Split in energy levels due to interaction between electron up or down spin and the electron orbitals.
Numerically explained by the Dirac equation when solving it for the hydrogen atom, and it is one of the main triumphs of the theory.
Small splits present in all levels due to interaction between the electron spin and the nuclear spin if it is present, i.e. the nucleus has an even number of nucleons.
As the name suggests, this energy split is very small, since the influence of the nucleus spin on the electron spin is relatively small compared to other fine structure.
TODO confirm: does it need quantum electrodynamics or is the Dirac equation enough?
The most important examples:
21 cm is very long and very low energy, because he energy split is very small!
Compare it e.g. with the hydrogen 1-2 spectral line which is 121.6 nm!
Split in the spectral line when a magnetic field is applied.
Non-anomalous: number of splits matches predictions of the Schrödinger equation about the number of possible states with a given angular momentum. TODO does it make numerical predictions?
Anomalous: evidence of spin.
www.pas.rochester.edu/~blackman/ast104/zeeman-split.html contains the hello world that everyone should know: 2p splits into 3 energy levels, so you see 3 spectral lines from 1s to 2p rather than just one.
p splits into 3, d into 5, f into 7 and so on, i.e. one for each possible azimuthal quantum number.
It also mentions that polarization effects become visible from this: each line is polarized in a different way. TODO more details as in an experiment to observe this.
Well explained at: Video 17. "Quantum Mechanics 7a - Angular Momentum I by ViaScience (2013)".
Video 1. Experimental physics - IV: 22 - Zeeman effect by Lehrportal Uni Gottingen (2020) Source.
This one is decent. Uses a cadmium lamp and an etalon on an optical table. They see a more or less clear 3-split in a circular interference pattern,
They filter out all but the transition of interest.
Video 2. Zeeman Effect - Control light with magnetic fields by Applied Science (2018) Source. Does not appear to achieve a crystal clear split unfortunately.
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.
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:
electron experiments: single electron double slit experiment.
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!
Video 3. 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.
Video 4. Electron Interference by the Italian National Research Council (1976) Source.
Institutional video about the 1974 single electron experiment by Merli, Missiroli, Pozzi from the University of Bologna.
Uses an electron biprism as in electron holography inside a transmission electron microscope.
Shows them manually making the biprism by drawing a fine glass wire and coating it with gold.
Then actually show the result live on a television screen, where you see the interference patterns only at higher electron currents, and then on photographic film.
This was elected "the most beautiful experiment" by readers of Physics World in 2002.
Accompanying website: l-esperimento-piu-bello-della-fisica.bo.imm.cnr.it/english/index.html.
Italian title: "Interferenza di elettroni". Goddammit, those Italian cinematographers can make even physics look exciting!
physics.stackexchange.com/questions/443358/in-the-double-slit-experiment-why-is-it-never-shown-that-particles-may-hit-the/573455#573455
It would be amazing to answer this with single particle double slit experiment measurements!
Quantum version of the Hall effect.
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 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:
  • Planck's on the discovery of the quantum theory (1900);
  • Einstein's on the light-quantum (1905);
  • Bohr's on the hydrogen atom (1913);
  • Bose's on what came to be called quantum statistics (1924);
  • Heisenberg's on what came to be known as matrix mechanics (1925);
  • and Schroedinger's on wave mechanics (1926).
Bibliography:

The Quantum Story by Jim Baggott (2011)

split words: 3
archive.org/details/quantumstoryhist0000bagg on the Internet Archive Open Library.
phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quantum_Mechanics_(Fitzpatrick)
This LibreTexts book does have some interest!
www.youtube.com/playlist?list=PLUl4u3cNGP60Zcz8LnCDFI8RPqRhJbb4L
100 10-20 minute videos properly split by topic, good resource!
Instructor: Barton Zwiebach.
Free material from university courses:
Author: David Tong.
Summary:
Looks very impressive! Last update marked 2011 as of 2020.
Goes up to "A.15 quantum field theory in a Nanoshell", Ciro have to review it to see if there's anything worthwhile in that section.
Personal page says he retired as of 2020: www.eng.fsu.edu/~dommelen/ But hopefully he has more time for these notes!
And he appears to have his own lightweight markup language that transpiles to LaTeX called l2h: www.eng.fsu.edu/~dommelen/l2h/
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.
These are the key mathematical ideas to understand!!
There are actually a few formulations out there. By far the dominant one as of 2020 has been the Schrödinger picture, which contrasts notably with the Heisenberg picture.
Another well known one is the de Broglie-Bohm theory, which is deterministic, but non-local.
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.
The state of a quantum system is a vector of unit length in a Hilbert space. TODO why Hilbert Space.
"Making a measurement" for an observable means applying a self-adjoint operator to the state, and after a measurement is done:
  • 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 7. "Visualization of Quantum Physics (Quantum Mechanics) by udiprod (2017)"
Those last two rules are also known as the Born rule.
Self adjoint operators are chosen because they have the following key properties:
  • their eigenvalues form an orthonormal basis
  • they are diagonalizable
See also: en.wikipedia.org/wiki/Measurement_in_quantum_mechanics
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 .
TODO: use the results from the quantum harmonic oscillator solution to precisely illustrate the discussion at Schrödinger picture with a concrete example.
Similar to quantum jump in the Bohr model, but for the Schrödinger equation.
The idea the the wave function of a small observed system collapses "obviously" cannot be the full physical truth, only a very useful approximation of reality.
Because then are are hard pressed to determine the boundary between what collapses and what doesn't, and there isn't such a boundary, as everything is interacting, including the observer.
The many-worlds interpretation is an elegant explanation for this. Though it does feel a bit sad and superfluous.
One single universal wavefunction, and every possible outcomes happens in some alternate universe. Does feel a bit sad and superfluous, but also does give some sense to perceived "wave function collapse".
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 .
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 along the 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 that is an operator and 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 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 ... 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.
This is basically what became the dominant formulation as of 2020 (and much earlier), and so we just call it the "mathematical formulation of quantum mechanics".
An "alternative" formulation of quantum mechanics that does not involve operators.
The first quantum mechanics theories developed.
Their most popular formulation has been the Schrödinger equation.
The partial differential equation of non-relativistic quantum mechanics.
Experiments explained:
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:
Equation 7. Schrodinger equation
where:
  • is the reduced Planck constant
  • is the wave function
  • is the time
  • is a linear operator called the Hamiltonian. It takes as input a function , 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 argument of could be anything, e.g.:
Note however that there is always a single magical 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.
Existence and uniqueness: mathoverflow.net/questions/212913/existence-and-uniqueness-for-two-dimensional-time-dependent-schr%C3%B6dinger-equation
The cool thing about the Time-independent Schrödinger equation is that we can always reduce solving the full Schrödinger equation to solving this slightly simpler time-independent version.
This is actually the approach that we will always take when solving the schrodinger equation, see e.g. quantum harmonic oscillator.
Before reading any further, you must understand heat equation solution with Fourier series, which uses separation of variables.
Once that example is clear, we see that the exact same separation of variables can be done to the Schrödinger equation. If we name the constant of the separation of variables for energy, we get::
  • a time-only part that does not depend on space and does not depend on the Hamiltonian at all. The solution for this part is therefore always the same exponentials for any problem, and this part is therefore "boring":
  • a space-only part that does not depend on time, bud does depend on the hamiltonian:
    Equation 9. Time-independent Schrodinger equation
    Since this is the only non-trivial part, unlike the time part which is trivial, this spacial part is just called "the time-independent Schrodinger equation".
    Note that the here is not the same as the in the time-dependent Schrodinger equation of course, as that psi is the result of the multiplication of the time and space parts. This is a bit of imprecise terminology, but hey, physics.
Because the time part of the equation is always the same and always trivial to solve, all we have to do to actually solve the Schrodinger equation is to solve the time independent one, and then we can construct the full solution trivially.
Once we've solve the time-independent part, call it for each discrete energy as , we proceed exactly as we did in heat equation solution with Fourier series: we make a weighted sum over all possible to match the initial condition, which is analogous to the Fourier series in the case of the heat equation to reach a final full solution:
Equation 10. Solution of the Schrodinger equation in terms of the time-independent and time dependent parts
and this is a solution by selecting such that at time we match the intial condition:
The fact that this approximation of the initial condition is always possible from is mathematically proven by some version of the spectral theorem bsaed on the fact that The Schrodinger equation Hamiltonian has to be Hermitian and therefore behaves nicely.
It is interesting to note that solving the time-independent Schrodinger equation can also be seen exactly as an eigenvalue equation where:
The only difference from usual matrix eigenvectors is that we are now dealing with an infinite dimensional vector space.
Furthermore:
Bibliography:
Where derivation == "intuitive routes", since a "law of physics" cannot be derived, only observed right or wrong.
TODO also comment on why are complex numbers used in the Schrodinger equation?.
Some approaches:
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:
The Schrödinger equation Hamiltonian has to be a Hermitian so we will have only positive energies I think: quantumcomputing.stackexchange.com/questions/12113/why-does-a-hamiltonian-have-to-be-hermitian
en.wikipedia.org/wiki/List_of_quantum-mechanical_systems_with_analytical_solutions
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.
Video 5. Simulation of the time-dependent Schrodinger equation (JavaScript Animation) by Coding Physics (2019) Source.
Source code: github.com/CodingPhysics/Schroedinger. One dimensional potentials, non-interacting particles. The code is clean, graphics based on github.com/processing/p5.js, and all maths from scratch. Source organization and comments are typical of numerical code, the anonymous author is was likely a Fortran user in the past.
A potential change patch in sketch.js: 
-   potential:     x => 2E+4*Math.pow((4*x - 1)*(4*x - 3),2),
+ potential:     x => 4*Math.pow(x - 0.5, 2),
Video 6. Quantum Mechanics 5b - Schrödinger Equation II by ViaScience (2013) Source. 2D non-interacting particle in a box, description says using Scilab and points to source. Has a double slit simulation.
Video 7. Visualization of Quantum Physics (Quantum Mechanics) by udiprod (2017) Source. Closed source, but a fantastic visualization and explanation of a 1D free wave packet, including how measurement snaps position to the measured range, position and momentum space and the uncertainty principle.
This is basically how quantum computing was first theorized by Richard Feynman: quantum computers as experiments that are hard to predict outcomes.
TODO answer that: quantumcomputing.stackexchange.com/questions/5005/why-it-is-hard-to-simulate-a-quantum-device-by-a-classical-devices. A good answer would be with a more physical example of quantum entanglement, e.g. on a photonic quantum computer.
We select for the general Equation 7. "Schrodinger equation":
giving the full explicit partial differential equation:
Equation 12. Schrödinger equation for a one dimensional particle
The corresponding time-independent Schrödinger equation for this equation is:
Equation 13. time-independent Schrödinger equation for a one dimensional particle
Schrödinger equation for a one dimensional particle with .
This equation is a subcase of Equation 12. "Schrödinger equation for a one dimensional particle" with .
We get the time-independent Schrödinger equation by substituting this into Equation 13. "time-independent Schrödinger equation for a one dimensional particle":
Now, there are two ways to go about this.
The first is the stupid "here's a guess" + "hey this family of solutions forms a complete bases"! This is exactly how we solved the problem at Section "Solving partial differential equations with the Fourier series", except that now the complete basis are the Hermite functions.
The second is the much celebrated ladder operator method.
A quantum version of the LC circuit!
TODO are there experiments, or just theoretical?
Show up in the solution of the quantum harmonic oscillator after separation of variables leading into the time-independent Schrödinger equation, much like solving partial differential equations with the Fourier series.
I.e.: they are both:
Not the same as Hermite polynomials.
www.physics.udel.edu/~jim/PHYS424_17F/Class%20Notes/Class_5.pdf by James MacDonald shows it well.
The operators are a natural guess on the lines of "if p and x were integers".
And then we can prove the ladder properties easily.
The commutator appear in the middle of this analysis.
Examples:
  • flash memory uses quantum tunneling as the basis for setting and resetting bits
  • alpha decay is understood as a quantum tunneling effect in the nucleus
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.
Video 8. A Better Way To Picture Atoms by minutephysics (2021) Source. Renderings based on the exact Schrödinger equation solution for the hydrogen atom that depict wave function concentration by concentration of small balls, and angular momentum by how fast the balls rotate at each point. Mentions that the approach is inspired by de Broglie-Bohm theory.
In the case of the Schrödinger equation solution for the hydrogen atom, each orbital is one eigenvector of the solution.
Remember from time-independent Schrödinger equation that the final solution is just the weighted sum of the eigenvector decomposition of the initial state, analogously to solving partial differential equations with the Fourier series.
This is the table that you should have in mind to visualize them: en.wikipedia.org/w/index.php?title=Atomic_orbital&oldid=1022865014#Orbitals_table
Quantum numbers appear directly in the Schrödinger equation solution for the hydrogen atom.
However, it very cool that they are actually discovered before the Schrödinger equation, and are present in the Bohr model (principal quantum number) and the Bohr-Sommerfeld model (azimuthal quantum number and magnetic quantum number) of the atom. This must be because they observed direct effects of those numbers in some experiments. TODO which experiments.
E.g. The Quantum Story by Jim Baggott (2011) page 34 mentions:
As the various lines in the spectrum were identified with different quantum jumps between different orbits, it was soon discovered that not all the possible jumps were appearing. Some lines were missing. For some reason certain jumps were forbidden. An elaborate scheme of ‘selection rules’ was established by Bohr and Sommerfeld to account for those jumps that were allowed and those that were forbidden.
This refers to forbidden mechanism. TODO concrete example, ideally the first one to be noticed. How can you notice this if the energy depends only on the principal quantum number?
Video 9. Quantum Numbers, Atomic Orbitals, and Electron configurations by Professor Dave Explains (2015) Source. He does not say the key words "Eigenvalues of the Schrödinger equation" (Which solve it), but the summary of results is good enough.
Determines energy. This comes out directly from the resolution of the Schrödinger equation solution for the hydrogen atom where we have to set some arbitrary values of energy by separation of variables just like we have to set some arbitrary numbers when solving partial differential equations with the Fourier series. We then just happen to see that only certain integer values are possible to satisfy the equations.
Fixed total angular momentum.
The direction however is not specified by this number.
To determine the quantum angular momentum, we need the magnetic quantum number, which then selects which orbital exactly we are talking about.
Fixed quantum angular momentum in a given direction.
Can range between .
E.g. consider gallium which is 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p1:
  • the electrons in s-orbitals such as 1s, 2d, and 3d are , and so the only value for is 0
  • the electrons in p-orbitals such as 2p, 3p and 4p are , and so the possible values for are -1, 0 and 1
  • the electrons in d-orbitals such as 2d are , and so the possible values for are -2, -1, 0 and 1 and 2
The z component of the quantum angular momentum is simply:
so e.g. again for gallium:
  • s-orbitals: necessarily have 0 z angular momentum
  • p-orbitals: have either 0, or z angular momentum
Note that this direction is arbitrary, since for a fixed azimuthal quantum number (and therefore fixed total angular momentum), we can only know one direction for sure. is normally used by convention.
This notation is cool as it gives the spin quantum number, which is important e.g. when talking about hyperfine structure.
But it is a bit crap that the spin is not given simply as but rather mixes up both the azimuthal quantum number and spin. What is the reason?
Video 10. Spectroscopic notation by Andre K (2014) Source.
Bibliography:
TODO. Can't find it easily. Anyone?
This is closely linked to the Pauli exclusion principle.
What does a particle even mean, right? Especially in quantum field theory, where two electrons are just vibrations of a single electron field.
Another issue is that if we consider magnetism, things only make sense if we add special relativity, since Maxwell's equations require special relativity, so a non approximate solution for this will necessarily require full quantum electrodynamics.
As mentioned at lecture 1 youtube.com/watch?video=H3AFzbrqH68&t=555, relativistic quantum mechanical theories like the Dirac equation and Klein-Gordon equation make no sense for a "single particle": they must imply that particles can pop in out of existence.
Bibliography:
Just ignore the electron electron interactions.
Bibliography:
No closed form solution, but good approximation that can be calculated by hand with the Hartree-Fock method, see hartree-Fock method for the helium atom.
Bibliography:
Video 11. Quantum Chemistry 9.2 - Helium Atom Energy Approximations by TMP Chem (2016) Source. Video gives the actual numerical value of various methods, second order perturbation theory being very close. But it the says that in the following videos will only do Hartree-Fock method.
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.
Boring rule that says that less energetic atomic orbitals are filled first.
Much more interesting is actually determining that order, which the Madelung energy ordering rule is a reasonable approximation to.
We will sometimes just write them without superscript, as it saves typing and is useless.
chemistry.stackexchange.com/questions/152/why-is-the-2s-orbital-lower-in-energy-than-the-2p-orbital-when-the-electrons-in
The principal quantum number thing fully determining energy is only true for the hydrogen emission spectrum for which we can solve the Schrödinger equation explicitly.
For other atoms with more than one electron, the orbital names are just a very good approximation/perturbation, as we don't have an explicit solution. And the internal electrons do change energy levels.
Note however that due to the more complex effect of the Lamb shift from QED, there is actually a very small 2p/2s shift even in hydrogen.
Looking at the energy level of the Schrödinger equation solution for the hydrogen atom, you would guess that for multi-electron atoms that only the principal quantum number would matter, azimuthal quantum number getting filled randomly.
However, orbitals energies for large atoms don't increase in energy like those of hydrogen due to electron-electron interactions.
In particular, the following would not be naively expected:
  • 2s fills up before 2p. From the hydrogen solution, you might guess that they would randomly go into either one as they'd have the same energy
  • in potassium fills up before 3d, even though it has a higher principal quantum number!
This rule is only an approximation, there exist exceptions to the Madelung energy ordering rule.
Bibliography:
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?
This notation is so confusing! People often don't manage to explain the intuition behind it, why this is an useful notation. When you see Indian university entry exam level memorization classes about this, it makes you want to cry.
The key reason why term symbols matter are Hund's rules, which allow us to predict with some accuracy which electron configurations of those states has more energy than the other.
web.chem.ucsb.edu/~devries/chem218/Term%20symbols.pdf puts it well: electron configuration notation is not specific enough, as each such notation e.g. 1s2 2s2 2p2 contains several options of spins and z angular momentum. And those affect energy.
This is why those symbols are often used when talking about energy differences: they specify more precisely which levels you are talking about.
Basically, each term symbol appears to represent a group of possible electron configurations with a given quantum angular momentum.
We first fix the energy level by saying at which orbital each electron can be (hyperfine structure is ignored). It doesn't even have to be the ground state: we can make some electrons excited at will.
The best thing to learn this is likely to draw out all the possible configurations explicitly, and then understand what is the term symbol for each possible configuration, see e.g. term symbols for carbon ground state.
It also confusing how uppercase letters S, P and D are used, when they do not refer to orbitals s, p and d, but rather to states which have the same angular momentum as individual electrons in those states.
It is also very confusing how extremelly close it looks to spectroscopic notation!
The form of the term symbol is:
The can be understood directly as the degeneracy, how many configurations we have in that state.
Video 12. Atomic Term Symbols by TMP Chem (2015) Source.
Video 13. Atomic Term Symbols by T. Daniel Crawford (2016) Source.
Bibliography:
Allow us to determine with good approximation in a multi-electron atom which electron configuration have more energy. It is a bit like the Aufbau principle, but at a finer resolution.
Note that this is not trivial since there is no explicit solution to the Schrödinger equation for multi-electron atoms like there is for hydrogen.
For example, consider carbon which has electron configuration 1s2 2s2 2p2.
If we were to populate the 3 p-orbitals with two electrons with spins either up or down, which has more energy? E.g. of the following two: 
m_L -1  0  1
    u_ u_ __
    u_ __ u_
    __ ud __
Higher spin multiplicity means lower energy. I.e.: you want to keep all spins pointin in the same direction.
This example covered for example at Video 14. "Term Symbols Example 1 by TMP Chem (2015)".
Carbon has electronic structure 1s2 2s2 2p2.
For term symbols we only care about unfilled layers, because in every filled layer the total z angular momentum is 0, as one electron necessarily cancels out each other:
  • magnetic quantum number varies from -l to +l, each with z angular momentum to and so each cancels the other out
  • spin quantum number is either + or minus half, and so each pair of electron cancels the other out
So in this case, we only care about the 2 electrons in 2p2. Let's list out all possible ways in which the 2p2 electrons can be.
There are 3 p orbitals, with three different magnetic quantum numbers, each representing a different possible z quantum angular momentum.
We are going to distribute 2 electrons with 2 different spins across them. All the possible distributions that don't violate the Pauli exclusion principle are:
m_l  +1  0 -1  m_L  m_S
     u_ u_ __    1    1
     u_ __ u_    0    1
     __ u_ u_   -1    1
     d_ d_ __    1   -1
     d_ __ d_    0   -1
     __ d_ d_   -1   -1
     u_ d_ __    1    0
     d_ u_ __    1    0
     u_ __ d_    0    0
     d_ __ u_    0    0
     __ u_ d_   -1    0
     __ d_ u_   -1    0
     ud __ __    2    0
     __ ud __    0    0
     __ __ ud   -2    0
where:
  • m_l is , the magnetic quantum number of each electron. Remember that this gives a orbital (non-spin) quantum angular momentum of to each such electron
  • m_L with a capital L is the sum of the of each electron
  • m_S with a capital S is the sum of the spin angular momentum of each electron
For example, on the first line: 
m_l  +1  0 -1  m_L  m_S
     u_ u_ __    1    1
we have:
  • one electron with , and so angular momentum
  • one electron with , and so angular momentum 0
and so the sum of them has angular momentum . So the value of is 1, we just omit the .
TODO now I don't understand the logic behind the next steps... I understand how to mechanically do them, but what do they mean? Can you determine the term symbol for individual microstates at all? Or do you have to group them to get the answer? Since there are multiple choices in some steps, it appears that you can't assign a specific term symbol to an individual microstate. And it has something to do with the Slater determinant. The previous lecture mentions it: www.youtube.com/watch?v=7_8n1TS-8Y0 more precisely youtu.be/7_8n1TS-8Y0?t=2268 about carbon.
youtu.be/DAgEmLWpYjs?t=2675 mentions that is not allowed because it would imply , which would be a state uu __ __ which violates the Pauli exclusion principle, and so was not listed on our list of 15 states.
He then goes for and mentions:
  • S = 1 so can only be 0
  • L = 2 (D) so ranges in -2, -1, 0, 1, 2
and so that corresponds to states on our list: 
ud __ __    2    0
u_ d_ __    1    0
u_ __ d_    0    0
__ u_ d_   -1    0
__ __ ud   -2    0
Note that for some we had a two choices, so we just pick any one of them and tick them off off from the table, which now looks like: 
 +1  0 -1  m_L  m_S
 u_ u_ __    1    1
 u_ __ u_    0    1
 __ u_ u_   -1    1
 d_ d_ __    1   -1
 d_ __ d_    0   -1
 __ d_ d_   -1   -1
 d_ u_ __    1    0
 d_ __ u_    0    0
 __ d_ u_   -1    0
 __ ud __    0    0
Then for the choices are:
  • S = 2 so is either -1, 0 or 1
  • L = 1 (P) so ranges in -1, 0, 1
so we have 9 possibilities for both together. We again verify that 9 such states are left matching those criteria, and tick them off, and so on.
For the , we have two electrons with spin up. The angular momentum of each electron is , and so given that we have two, the total is , so again we omit and is 1.
Video 14. Term Symbols Example 1 by TMP Chem (2015) Source. Carbon atom.
Bibliography:
Can we make any ab initio predictions about it all?
A 2016 paper: aip.scitation.org/doi/abs/10.1063/1.4948309
Video 15. Quantum Chemistry 10.1 - Hydrogen Molecule Hamiltonian by TMP Chem (2016) Source. Continued in the following videos, uses the Born–Oppenheimer approximation. Does not give predictions vs experiment?
Isomers were quite confusing for early chemists, before atomic theory was widely accepted, and people where thinking mostly in terms of proportions of equations, related: Section "Isomers suggest that atoms exist (1874)".
Exist because double bonds don't rotate freely. Have different properties of course, unlike enantiomer.
Bibliography:
Mirror images.
Key exmaple: d and L amino acids. Enantiomers have identical physico-chemical properties. But their biological roles can be very different, because an enzyme might only be able to act on one of them.
TODO definition. Appears to be isomers
Example:
Molecules that are the same if you just look at "what atom is linked to what atom", they are only different if you consider the relative spacial positions of atoms.
Discrete quantum system model that can model both spin in the Stern-Gerlach experiment or photon polarization in polarizer.
Also known in quantum computing as a qubit :-)
physics.stackexchange.com/questions/204090/understanding-the-bloch-sphere/598254#598254
A more concrete and easier to understand version of it is the more photon-specific Poincaré sphere, have a look at that one first.
2D representation of .
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 7. "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 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:
  • to localize the wave in space at position 0 to reduce the space uncertainty, we have to reduce the support. However, doing so makes the momentum variation on the edges more and more important, as the slope will go up and down faster (higher top, and less x space for descent), leading to a larger variance (note that average momentum is still 0, due to to symmetry of the bump function)
  • to localize the momentum as much as possible at 0, we can make the support wider and wider. This makes the bumps at the edges smaller and smaller. However, this also obviously delocalises the wave function more and more, increasing the variance of x
Bibliography:
One of the main reasons why physicists are obsessed by this topic is that position and momentum are mapped to the phase space coordinates of Hamiltonian mechanics, which appear in the matrix mechanics formulation of quantum mechanics, which offers insight into the theory, particularly when generalizing to relativistic quantum mechanics.
One way to think is: what is the definition of space space? It is a way to write the wave function such that:
  • the position operator is the multiplication by
  • the momentum operator is the derivative by
And then, what is the definition of momentum space? It is of course a way to write the wave function such that:
  • the momentum operator is the multiplication by
physics.stackexchange.com/questions/39442/intuitive-explanation-of-why-momentum-is-the-fourier-transform-variable-of-posit/39508#39508 gives the best idea intuitive idea: the Fourier transform writes a function as a (continuous) sum of plane waves, and each plane wave has a fixed momentum.
Bibliography:
A way to write the wavefunction such that the position operator is:
i.e., a function that takes the wavefunction as input, and outputs another function:
If you believe that mathematicians took care of continuous spectrum for us and that everything just works, the most concrete and direct thing that this representation tells us is that:
the probability of finding a particle between and at time
equals:
This operator case is surprisingly not necessarily mathematically trivial to describe formally because you often end up getting into the Dirac delta functions/continuous spectrum: as mentioned at: mathematical formulation of quantum mechanics
One dimension in position representation:
In three dimensions In position representation, we define it by using the gradient, and so we see that
Video 16. Position and Momentum from Wavefunctions by Faculty of Khan (2018) Source. Proves in detail why the momentum operator is . The starting point is determining the time derivative of the expectation value of the position operator.
Appears directly on Schrödinger equation! And in particular in the time-independent Schrödinger equation.
There is also a time-energy uncertainty principle, because those two operators are also complementary.
Basically the operators are just analogous to the classical ones e.g. the classical:
becomes:
Besides the angular momentum in each direction, we also have the total angular momentum:
Then you have to understand what each one of those does to the each atomic orbital:
There is an uncertainty principle between the x, y and z angular momentums, we can only measure one of them with certainty at a time. Video 17. "Quantum Mechanics 7a - Angular Momentum I by ViaScience (2013)" justifies this intuitively by mentioning that this is analogous to precession: if you try to measure electrons e.g. with the Zeeman effect the precess on the other directions which you end up modifing.
Video 17. Quantum Mechanics 7a - Angular Momentum I by ViaScience (2013) Source.
See: angular momentum operator.
physics.stackexchange.com/questions/229885/energy-equation-in-quantum-mechanics
TODO is there any good intuitive argument or proof of conservation of energy, momentum, angular momentum?
Proof that the probability 1 is conserved by the time evolution:
It can be derived directly from the Schrödinger equation.
Bibliography:
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 7. "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 5. "Simulation of the time-dependent Schrodinger equation (JavaScript Animation) by Coding Physics (2019)".
www.nature.com/articles/119890a0
Relates particle momentum and its wavelength, or equivalently, energy and frequency.
The wavelength relation is:
but since:
the wavelength relation implies:
Particle wavelength can be for example measured very directly on a double-slit experiment.
So if we take for example electrons of different speeds, we should be able to see the diffraction pattern change accordingly.
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.
This Heisenberg's breakthrough paper on matrix mechanics which later led to the Schrödinger equation, see also: history of quantum mechanics.
Published on the Zeitschrift für Physik volume 33 page pages 879-893, link.springer.com/article/10.1007%2FBF01328377
Modern overview: www.mat.unimi.it/users/galgani/arch/heisenberg25amer_j_phys.pdf
Basically the same as matrix mechanics it seems, just a bit more generalized.
Deterministic, but non-local.
Photon energy is proportional to its frequency:
or with common weird variables:
This only makes sense if the photon exists, there is no classical analogue, because the energy of classical waves depends only on their amplitude, not frequency.
Experiments that suggest this:
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 , 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:
Appears in the Schrödinger equation.
Equals the quantum of angular momentum in the Bohr model.
This section is present in another page, follow this link to view it.
Quantum field theory lecture by Tobias Osborne (2017) mentions that quantization is a guess.
This is one of the first examples in most quantum field theory.
It usually does not involve any forces, just the interpretation of what the quantum field is.
www.youtube.com/watch?v=zv94slY6WqY&list=PLSpklniGdSfSsk7BSZjONcfhRGKNa2uou&index=2 Quantization Of A Free Real Scalar Field by Dietterich Labs (2019)
Quantum superposition is really weird because it is fundamentally different than "either definite state but I don't know which", because the superposition state leads to different measurements than the non-superposition state.
Examples:
Quantum entanglement is often called spooky/surprising/unintuitive, but they key question is to understand why.
To understand that, you have to understand why it is fundamentally impossible for the entangled particle pair be in a predefined state according to experiments done e.g. where one is deterministically yes and the other deterministically down.
In other words, why local hidden-variable theory is not valid.
How to generate entangled particles:
Video 18. Bell's Theorem: The Quantum Venn Diagram Paradox by minutephysics (2017) Source.
Contains the clearest Bell test experiment description seen so far.
It clearly describes the photon-based 22.5, 45 degree/85%/15% probability photon polarization experiment and its result conceptually.
It does not mention spontaneous parametric down-conversion but that's what they likely hint at.
Done in Collaboration with 3Blue1Brown.
Question asking further clarification on why the 100/85/50 thing is surprising: physics.stackexchange.com/questions/357039/why-is-the-quantum-venn-diagram-paradox-considered-a-paradox/597982#597982
Video 19. Bell's Inequality I by ViaScience (2014) Source.
Video 20. Quantum Entanglement & Spooky Action at a Distance by Veritasium (2015) Source. Gives a clear explanation of a thought Bell test experiments with electron spin of electron pairs from photon decay with three 120-degree separated slits. The downside is that he does not clearly describe an experimental setup, it is quite generic.
Video 21. Quantum Mechanics: Animation explaining quantum physics by Physics Videos by Eugene Khutoryansky (2013) Source. Usual Eugene, good animations, and not too precise explanations :-) youtu.be/iVpXrbZ4bnU?t=922 describes a conceptual spin entangled electron-positron pair production Stern-Gerlach experiment as a Bell test experiments. The 85% is mentioned, but not explained at all.
Video 22. Quantum Entanglement: Spooky Action at a Distance by Don Lincoln (2020) Source. This only has two merits compared to Video 20. "Quantum Entanglement & Spooky Action at a Distance by Veritasium (2015)": it mentions the Aspect et al. (1982) Bell test experiment, and it shows the continuous curve similar to en.wikipedia.org/wiki/File:Bell.svg. But it just does not clearly explain the bell test.
Video 23. Quantum Entanglement Lab by Scientific American (2013) Source. The hosts interview Professor Enrique Galvez of Colgate University who shows briefly the optical table setup without great details, and then moves to a whiteboard explanation. Treats the audience as stupid, doesn't say the keywords spontaneous parametric down-conversion and Bell's theorem which they clearly allude to. You can even them showing a two second footage of the professor explaining the rotation experiments and the data for it, but that's all you get.
Basically a precise statement of "quantum entanglement is spooky".
Some of the most remarkable ones seem to be:
  • Alain Aspect 1982
  • Hensen et al., Giustina et al., Shalm et al. (2015): "loophole-free" Bell tests

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