Ciro Santilli @cirosantilli  40

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 $2$ 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?

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Forbidden mechanism Updated 2025-07-16

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Bibliography:

phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quantum_Mechanics_(Fitzpatrick)/12%3A_Time-Dependent_Perturbation_Theory/12.13%3A_Forbidden_Transitions from Introductory Quantum Mechanics by Richard Fitzpatrick (2020)

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Rydberg atom Updated 2025-07-16

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Hydrogen emission spectrum Updated 2025-07-16

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Fine structure Updated 2025-07-16

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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.

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Zeeman effect Updated 2025-07-16

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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 "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.

youtu.be/ZmObNFAqkBE?t=165 passes the lines through a polarizer, which shows how orbital angular momentum is carried by photon polarization
youtu.be/ZmObNFAqkBE?t=370 says they are looking at 1D2 to 1P1 changes.

Video 2.

Zeeman Effect - Control light with magnetic fields by Applied Science (2018)

Source. Does not appear to achieve a crystal clear split unfortunately.

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Synthetic chromosome Updated 2025-07-16

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Basically a synonym for doing a large chunk of de novo DNA synthesis.

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Synthetic virus Updated 2025-11-18

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Man-made virus!

TODO: if we had cheap de novo DNA synthesis, how hard would it be to bootstrap a virus culture from that? github.com/cirosantilli/cirosantilli.github.io/issues/60

Is it easy to transfect a cell with the synthesized DNA, and get it to generate full infectious viral particles?

If so, then de novo DNA synthesis would be very similar to 3D printed guns: en.wikipedia.org/wiki/3D_printed_firearms.

It might already be possible to order dissimulated sequences online:

www.theguardian.com/world/2006/jun/14/terrorism.topstories3

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Genome Project-Write Updated 2025-07-16

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Yeast artificial chromosome Updated 2025-07-16

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Special unitary of degree 2 Updated 2025-07-16

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en.wikipedia.org/wiki/Representation_theory_of_SU(2)

Double cover of

SO (3)

Isomorphic to the quaternions.

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Light cone Updated 2025-07-16

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A subset of Spacetime diagram.

The key insights that it gives are:

future and past are well defined: every reference frame sees your future in your future cone, and your past in your past cone
Otherwise causality could be violated, and then things would go really bad, you could tell your past self to tell your past self to tell your past self to do something.
You can only affect the outcome of events in your future cone, and you can only be affected by events in your past cone. You can't travel fast enough to affect.
Two spacetime events with such fixed causality are called timelike-separated events.
every other event (to right and left, known as spacelike-separated events) can be measured to happen before or after your current spacetime event by different observers.
But that does not violate causality, because you just can't reach those spacetime points anyways to affect them.

Figure 1.
Animation showing how space-separated events can be observed to happen in different orders by observers in different frames of reference
. Source.

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String-sorting algorithm Updated 2025-07-16

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Schrödinger equation for a one dimensional particle Updated 2025-07-16

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We select for the general Equation "Schrodinger equation":

$x = x$ , the linear cartesian coordinate in the x direction
$\hat{H} = - \frac{ℏ ^{2}}{2 m} \frac{\partial ^{2}}{\partial x ^{2}} + V (x, t)$ , which analogous to the sum of kinetic and potential energy in classical mechanics

giving the full explicit partial differential equation:

i ℏ \frac{\partial ψ ( x , t )}{\partial t} = [- \frac{ℏ ^{2}}{2 m} \frac{\partial ^{2}}{\partial x ^{2}} + V (x, t)] ψ (x, t)

(1)

Equation 1.

Schrödinger equation for a one dimensional particle

The corresponding time-independent Schrödinger equation for this equation is:

[- \frac{ℏ ^{2}}{2 m} \frac{\partial ^{2}}{\partial x ^{2}} + V (x)] ψ (x) = E ψ (x)

(2)

Equation 2.

time-independent Schrödinger equation for a one dimensional particle

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Schrödinger equation for a free one dimensional particle Updated 2025-07-16

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Schrödinger equation for a one dimensional particle with

V = 0

. The first step is to calculate the time-independent Schrödinger equation for a free one dimensional particle

Then, for each energy

E

, from the discussion at Section "Solving the Schrodinger equation with the time-independent Schrödinger equation", the solution is:

ψ (x) = \int_{E = - \infty}^{\infty} e^{\frac{2 m E i x}{ℏ}} e^{- i Et /ℏ} = e^{i \frac{2 m E x - Et}{ℏ}}

(1)

Therefore, we see that the solution is made up of infinitely many plane wave functions.

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Particle in a box Updated 2025-07-16

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Quantum harmonic oscillator Updated 2025-07-16

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This equation is a subcase of Equation "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 "time-independent Schrödinger equation for a one dimensional particle":

[- \frac{ℏ}{2 m} \frac{\partial ^{2}}{\partial x} + x^{2}] ψ = E ψ (x)

(1)

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.

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Quantum tunnelling Updated 2025-07-16

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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

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Schrödinger equation solution for the hydrogen atom Updated 2025-07-16

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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.

Video 1.

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.

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Solutions for the Schrodinger equation with multiple particles Updated 2025-07-16

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Bibliography:

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

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 There are unlisted articles, also show them or only show them.