Quantum mechanics is quite a broad term. Perhaps it is best to start approaching it from the division into:

- non-relativistic quantum mechanics: obviously the simpler one, and where you should start
- relativistic quantum mechanics: more advanced, and arguably "less useful"

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: https://physics.stackexchange.com/questions/20003/why-dont-electrons-crash-into-the-nuclei-they-orbit

In other sections:

- black-body radiation experiment
- Einstein solid experiments, which are analogous to black body radiation experiments

- emission spectrum
- electron diffraction experiments such as:

Bibliography:

- http://web.mit.edu/course/5/5.73/oldwww/Fall04/notes/Experimental_Evidence_for_Quantum_Mechanics.pdf Experimental Evidence for Quantum Mechanics

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:

- Schrödinger equation: major lines predicted, including Zeeman effect, but not finer line splits like fine structure
- Dirac equation: explains fine structure 2p spin split due to electron spin/orbit interactions, but not Lamb shift
- quantum electrodynamics: explains Lamb shift
- hyperfine structure: due to electron/nucleus spin interactions, offers a window into nuclear spin

Let's do a sanity check.

Searching for "H" for hydrogen leads to: https://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 $2$ 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?

One reasonable and memorable approximation excluding any fine structure is:
see for example example:
hydrogen 1-2 spectral line.

$E_{n}=−n_{2}13.6eV $

Equation 1. "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 ($2.47×10_{1}5$ Hz), which is a reasonable match with the value of 121.567... from the NIST Atomic Spectra Database.

$E_{n}=−13.6eV[2_{2}1 −1_{2}1 ]=10.2eV$

Kind of a synonym for hydrogen spectral series, 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.

For example, for the hydrogen emission spectrum:

- hydrogen 1-2 spectral line: 10.2eV

TODO confirm: does it need quantum electrodynamics or is the Dirac equation enough?

Notable application: caesium standard, which is used to define the second in the International System of Units since 1967.

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?

http://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 12. "Quantum Mechanics 7a - Angular Momentum I by ViaScience (2013)".

Amazingly confirms the wave particle duality of quantum mechanics.

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: https://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:

- https://aapt.scitation.org/doi/full/10.1119/1.4955173 "Video recording true single-photon double-slit interference" by Aspden and Padgetta (2016). Abstract says using spontaneous parametric down-conversion detection of the second photon to know when to turn the camera on

electron experiments: single electron double slit experiment.

Non-elementary particle:

- 2019-10-08: 25,000 Daltons
- https://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!

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:

- https://physics.stackexchange.com/questions/18632/good-book-on-the-history-of-quantum-mechanics on Physics Stack Exchange
- https://www.youtube.com/watch?v=5hVmeOCJjOU A Brief History of Quantum Mechanics by Sean Carroll (2020) Given at the Royal Institution.

- 1859-1900: see Section "Black-body radiation experiment". Continuously improving culminating in Planck's law black-body radiation and Planck's law
- 1905 photoelectric effect and the photon
- TODO experiments
- 1905 Einstein's photoelectric effect paper. Planck was intially thinking that light was continuous, but the atoms vibrated in a discrete way. Einstein's explanation of the photoelectric effect throws that out of the window, and considers the photon discrete.

- 1913 atomic spectra and the Bohr model
- 1885 Balmer series, an empirical formula describes some of the lines of the hydrogen emission spectrum
- 1888 Rydberg formula generalizes the Balmer series
- 1896 Pickering series makes it look like a star has some new kind of hydrogen that produces half-integer entries in the Pickering series
- 1911 Bohr visits J. J. Thomson in the University of Cambridge for his postdoc, but they don't get along well
- Bohr visits Rutherford at the University of Manchester and decides to transfer there. During this stay he becomes interested in problems of the electronic structure of the atom.Bohr was forced into a quantization postulate because spinning electrons must radiate energy and collapse, so he postulated that electrons must somehow magically stay in orbits without classically spinning.

- 1913 february: young physics professor Hans Hansen tells Bohr about the Balmer series. This is one of the final elements Bohr needed.
- 1913 Bohr model published predicts atomic spectral lines in terms of the Planck constant and other physical constant.
- explains the Pickering series as belonging to inoized helium that has a single electron. The half term in the spectral lines of this species come from the nucleus having twice the charge of hydrogen.
- 1913 March: during review before publication, Rutherford points out that instantaneous quantum jumps don't seem to play well with causality.

- 1916 Bohr-Sommerfeld model introduces angular momentum to explain why some lines are not observed, as they would violate the conservation of angular momentum.

100 10-20 minute videos properly split by topic, good resource!

Instructor: Barton Zwiebach.

Free material from university courses:

- https://physics.weber.edu/schroeder/quantum/QuantumBook.pdf (archive) "Notes on Quantum Mechanics" pusbliehd by Daniel V. Schroeder (2019) The author is from from Weber State University.

Author: David Tong.

Summary:

- Chapter 2 "Band Structure" covers electronic band theory

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: http://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: http://www.eng.fsu.edu/~dommelen/l2h/

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.

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: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 7. "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 $ϕ$.

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.

$(⟨x∣)∣y⟩$

$x⋅y$

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 $x⋅y$ 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.

$⟨x∣+⟨y∣$

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 $v$... 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.

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".

- https://physics.stackexchange.com/questions/1201/linearity-of-quantum-mechanics-and-nonlinearity-of-macroscopic-physics
- https://physics.stackexchange.com/questions/134503/what-is-the-physical-reason-behind-linearity-of-schrodingers-equation
- https://physics.stackexchange.com/questions/33344/is-the-universe-linear-if-so-why

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.

Experiments explained:

- via the Schrödinger equation solution for the hydrogen atom it predicts:
- spectral line basic lines, plus Zeeman effect

- Schrödinger equation solution for the helium atom: perturbative solutions give good approximations to the energy levels
- double-slit experiment: I think we have a closed solution for the max and min probabilities on the measurement wall, and they match experiments

Experiments not explained: those that the Dirac equation explains like:

- fine structure
- spontaneous emission coefficients

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:
where:

$iℏ∂t∂ψ(x∣,t) =H^[ψ(x∣,t)]$

- $ℏ$ is the reduced Planck constant
- $ψ$ is the wave function
- $t$ is the time
- $H^$ is a linear operator called the Hamiltonian. It takes as input a function $ψ$, 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 $x∣$ argument of $ψ$ could be anything, e.g.:Note however that there is always a single magical $t$ 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.

- we could have preferred polar coordinates instead of linear ones if the potential were symmetric around a point
- we could have more than one particle, e.g. solutions of the Schrodinger equation for two electrons, which would have e.g. $x_{1}$ and $x_{2}$ for different particles. No matter how many particles there are, we have just a single $ψ$, we just add more arguments to it.
- we could have even more generalized coordinates. This is much in the spirit of Hamiltonian mechanics or generalized coordinates

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.

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 $E$ 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":
$e_{−iEt/ℏ}$
- a space-only part that does not depend on time, bud does depend on the hamiltonian:$H^[ψ(x∣)]=Eψx∣$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 $ψ$ here is not the same as the $ψ$ 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 $E_{i}$ as $E_{i}ψ_{i}(x)$, we proceed exactly as we did in heat equation solution with Fourier series: we make a weighted sum over all possible $E$ 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:
and this is a solution by selecting
$E_{i}$ such that at time $t=0$ we match the intial condition:

$∑_{i=0}e_{−iEt/ℏ}E_{i}ψ_{i}(x)$

$∑_{i=0}e_{−iE0/ℏ}E_{i}ψ_{i}(x)=∑_{i=0}E_{i}ψ_{i}(x)=initialcondition$

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.

- the Hamiltonian is a linear operator
- the value of the energy
`E`

is an eigenvalue

Furthermore:

- we immediately see from the equation that the time-independent solutions are states of deterministic energy because the energy is an eigenvalue of the Hamiltonian operator
- by looking at Equation 9. "Solution of the Schrodinger equation in terms of the time-independent and time dependent parts", it is obvious that if we take an energy measurement, the probability of each result never changes with time, because it is only multiplied by a constant

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:

- https://en.wikipedia.org/w/index.php?title=Schr%C3%B6dinger_equation&oldid=964460597#Derivation: holy crap, this just goes all in into a Lie group approach, nice
- Richard Feynman's derivation of the Schrodinger equation:
- https://physics.stackexchange.com/questions/263990/feynmans-derivation-of-the-schrödinger-equation
- https://www.youtube.com/watch?v=xQ1d0M19LsM "Class Y. Feynman's Derivation of the Schrödinger Equation" by doctorphys (2020)

- https://www.youtube.com/watch?v=zC_gYfAqjZY&list=PL54DF0652B30D99A4&index=53 "I5. Derivation of the Schrödinger Equation" by doctorphys

Why is there a complex number in the equation? Intuitively and mathematically:

- https://physics.stackexchange.com/questions/8062/about-the-complex-nature-of-the-wave-function
- https://physics.stackexchange.com/questions/32422/qm-without-complex-numbers
- https://physics.stackexchange.com/questions/292947/is-it-possible-to-formulate-the-schr%C3%B6dinger-equation-in-a-manner-that-excludes-i

Some ideas:

The Schrödinger equation Hamiltonian has to be a Hermitian so we will have only positive energies I think: https://quantumcomputing.stackexchange.com/questions/12113/why-does-a-hamiltonian-have-to-be-hermitian

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.

- https://www.youtube.com/watch?v=1Z9wo2CzJO8 "Schrodinger equation solved numerically in 3D" by Tetsuya Matsuno. 3D hydrogen atom, code may be hidden in some paper, maybe
- https://www.youtube.com/playlist?list=PLdCdV2GBGyXM0j66zrpDy2aMXr6cgrBJA "Computational Quantum Mechanics" by Let's Code Physics. Uses a 1D trinket.io.
- https://www.youtube.com/watch?v=BBt8EugN03Q Simulating Quantum Systems [Split Operator Method] by LeiosOS (2018)
- https://www.youtube.com/watch?v=86x0_-JGlGQ Simulating the Quantum World on a Classical Computer by Garnet Chan (2016) discusses how modeling only local entanglement can make certain simulations feasible

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: https://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 6. "Schrodinger equation":giving the full explicit partial differential equation:
The corresponding
time-independent Schrödinger equation for this equation is:

- $x∣=x$, the linear cartesian coordinate in the x direction
- $H^=−2mℏ_{2} ∂x_{2}∂_{2} +V(x,t)$, which analogous to the sum of kinetic and potential energy in classical mechanics

$iℏ∂t∂ψ(x,t) =[−2mℏ_{2} ∂x_{2}∂_{2} +V(x,t)]ψ(x,t)$

$[−2mℏ ∂x∂_{2} +V(x)]ψ=Eψ(x)$

This equation is a subcase of Equation 11. "Schrödinger equation for a one dimensional particle" with $V(x)=x_{2}$.

We get the time-independent Schrödinger equation by substituting this $V$ into Equation 12. "time-independent Schrödinger equation for a one dimensional particle":

$[−2mℏ ∂x∂_{2} +x_{2}]ψ=Eψ(x)$

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 basis"! 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.

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.

Not the same as Hermite polynomials.

http://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 major spectral line of the hydrogen atom by taking the difference between energy levels

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

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: https://en.wikipedia.org/w/index.php?title=Atomic_orbital&oldid=1022865014#Orbitals_table

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.

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. $z$ is normally used by convention.

Quantum Mechanics for Engineers by Leon van Dommelen (2011)

Bibliography:

- quantum Mechanics for Engineers by Leon van Dommelen (2011) "5. Multiple-Particle Systems"

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 https://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:

- https://www.youtube.com/watch?v=Og13-bSF9kA&list=PLDfPUNusx1Eo60qx3Od2KLUL4b7VDPo9F "Advanced quantum theory" by Tobias J. Osborne says that the course will essentially cover multi-particle quantum mechanics!
- https://physics.stackexchange.com/questions/54854/equivalence-between-qft-and-many-particle-qm "Equivalence between QFT and many-particle QM"
- Course: Quantum Many-Body Physics in Condensed Matter by Luis Gregorio Dias (2020) from course: Quantum Many-Body Physics in Condensed Matter by Luis Gregorio Dias (2020) give a good introduction to non-interacting particles

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.

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.

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.

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
- $4s_{1}$ 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.

Can we make any ab initio predictions about it all?

A 2016 paper: https://aip.scitation.org/doi/abs/10.1063/1.4948309

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.

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:

- the three most table polymorphs of calcium carbonate polymorphs are:

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 :-)

A more concrete and easier to understand version of it is the more photon-specific Poincaré sphere, have a look at that one first.

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 $L_{2}$ due to Carleson's theorem.

Then the uncertainty principle follows immediately from a general property of the Fourier transform: https://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:

- https://www.youtube.com/watch?v=bIIjIZBKgtI&list=PL54DF0652B30D99A4&index=59 "K2. Heisenberg Uncertainty Relation" by doctorphys (2011)
- https://physics.stackexchange.com/questions/132111/uncertainty-principle-intuition Uncertainty Principle Intuition on Physics Stack Exchange

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 $ψ_{x}(x)$ such that:And then, what is the definition of momentum space? It is of course a way to write the wave function $ψ_{p}(p)$ such that:

- the position operator is the multiplication by $x$
- the momentum operator is the derivative by $x$

- the momentum operator is the multiplication by $p$

https://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.

A way to write the wavefunction $ψ(x)$ such that the position operator is:
i.e., a function that takes the wavefunction as input, and outputs another function:

$x$

$xψ(x)$

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 $x_{0}$ and $x_{1}$ at time $t$equals:

$∫_{x_{0}}xψx,tdx$

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:

$p^ =−iℏ∂x∂ $

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.

Besides the angular momentum in each direction, we also have the total angular momentum:

$L^_{2}=L^_{x}+L^_{y}+L^_{z}$

Then you have to understand what each one of those does to the each atomic orbital:

- total angular momentum: determined by the azimuthal quantum number
- angular momentum in one direction ($z$ by convention): determined by the magnetic quantum number

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 12. "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.

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:

- https://quantummechanics.ucsd.edu/ph130a/130_notes/node127.html from quantum physics by Jim Branson (2003).That proof also mentions that if the potential
`V`

is not real, then there is no conservation of probability! Therefore the potential*must*be real valued!

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)".

Relates particle momentum and its wavelength, or equivalently, energy and frequency.

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, https://link.springer.com/article/10.1007%2FBF01328377

Basically the same as matrix mechanics it seems, just a bit more generalized.

Deterministic, but non-local.

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 $lim_{h→0}$, 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:

$6.62607015×10_{−34}J⋅s$

Appears in the Schrödinger equation.

Equals the quantum of angular momentum in the Bohr model.

quantum field theory lecture by Tobias Osborne (2017) mentions that quantization is a guess.

We make observations

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.

https://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:

- https://www.youtube.com/watch?v=tt8gVXDsh7Q "Interference in quantum mechanics" by Looking Glass Universe (2015) shows how a left-right spin measurement has a defined value for a superposed half up half down state, but not for a pure up state.TODO can this be conducted? As mentioned in the video, this is closely linked to the fact that you can describe the wave function in multiple different bases (up/down or left/right), which is also at the root of the uncertainty principle.
- Video "Quantum Mechanics 9b - Photon Spin and Schrodinger's Cat II by ViaScience (2013)" gives a similar photon version
- it seems that the single particle double slit experiment can also be thought of as in terms of a superposition of "the particle goes through the right" and "the particle goes through the right", although it is a bit harder to thing about as it is not a discrete process

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:

- particle decay, notably pair production
- for photons, notably: spontaneous parametric down-conversion, e.g.: https://www.youtube.com/watch?v=tn1sEaw1K2k "Shanni Prutchi Construction of an Entangled Photon Source" by HACKADAY (2015). Estimatd price: 5000 USD.

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

- Bra-ket notation
- Classical limit
- Classical mechanics
- Condensed matter physics
- Continuous spectrum
- Correspondence principle
- Derivation of the Klein-Gordon equation
- Dirac delta function
- Distribution
- Double-slit experiment
- Dual vector
- Einstein solid
- Entropy
- Half-life
- Hilbert space
- History of quantum mechanics
- Inward Bound by Abraham Pais (1988)
- Lebesgue integral of $L_{p}$ is complete but Riemann isn't
- Matrix mechanics
- Maxwell's equations in curved spacetime
- Maxwell's equations with pointlike particles
- Non-relativistic quantum mechanics
- GitHub
- Motivation
- Perturbation theory
- Phase-space formulation
- Position and momentum space
- Quantum computers as experiments that are hard to predict outcomes
- Quantum electrodynamics
- Quantum key distribution
- Real world applications of the Lebesgue integral
- Relativistic quantum mechanics
- Richard Feynman
- Schrödinger equation
- Spectral line
- Why you should give money to Ciro Santilli
- The Fourier transform is a bijection in $L_{2}$
- Tobias J. Osborne
- ViaScience
- Wheeler-Feynman absorber theory