Ciro Santilli @cirosantilli 34

 Incoming links: Fine structure

B3 Oxford physics course  Updated 2024-12-15  +Created 1970-01-01

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users.physics.ox.ac.uk/~lvovsky/B3/ contain assorted PDFs from between 2015 and 2019

Syllabus reads:

Multi-electron atoms: central field approximation, electron configurations, shell structure, residual electrostatic interaction, spin orbit coupling (fine structure).
Spectra and energy levels: Term symbols, selection rules, X-ray notation, Auger transitions.
Hyperfine structure; effects of magnetic fields on fine and hyperfine structure. Presumably Zeeman effect.
Two level system in a classical light field: Rabi oscillations and Ramsey fringes, decaying states; Einstein
A and B coefficients; homogeneous and inhomogeneous broadening of spectral lines; rate equations.
Optical absorption and gain: population inversion in 3- and 4-level systems; optical gain cross section; saturated absorption and gain.

Professor in 2000s seems to be

en.wikipedia.org/wiki/Paul_Ewart. He actually fought not to be dismissed by age and won!
www.physics.ox.ac.uk/our-people/ewart

But as of 2023 marked emeritus, so who took over?

Ewart is actually religious:

www.youtube.com/watch?v=aulL-Qa65i0 Paul Ewart, Chance, Science and Spirituality by Faraday Institute for Science and Religion. Oh, he is/was actually chairman of that crap
www.youtube.com/watch?v=PVX2F4XvGYo Chaos and the Character of God by Prof. Paul Ewart

This dude is pure trouble for Oxford!

Undated materials Ewart:

users.physics.ox.ac.uk/~ewart/index.htm
users.physics.ox.ac.uk/~ewart/Atomic%20Physics%20lecture%20notes%20C%20port.pdf
slides: users.physics.ox.ac.uk/~ewart/Atomic%20Physics%20Lecture%20PPT%20slides%201_8.pdf. Also under: www2.physics.ox.ac.uk/sites/default/files/2011-10-19/atomic_physics_lectures_1_8_09_pdf_pdf_18283.pdf. The course was previously B1, they just change the IDs randomly from time to time to fit the B1-7 numbering.

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Dirac equation  Updated 2024-12-15  +Created 1970-01-01

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Adds special relativity to the Schrödinger equation, and the following conclusions come basically as a direct consequence of this!

Experiments explained:

spontaneous emission coefficients.
fine structure, notably for example Dirac equation solution for the hydrogen atom
antimatter
particle creation and annihilation

Experiments not explained: those that quantum electrodynamics explains like:

Lamb shift
TODO: quantization of the electromagnetic field as photons?

The Dirac equation is a set of 4 partial differential equations on 4 complex valued wave functions. The full explicit form in Planck units is shown e.g. in Video 1. "Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)" at youtu.be/OCuaBmAzqek?t=1010:

i \partial_{t} ⎣ ⎢ ⎢ ⎢ ⎡ ψ_{1} ψ_{2} ψ_{3} ψ_{4} ⎦ ⎥ ⎥ ⎥ ⎤ = - i \partial_{x} ⎣ ⎢ ⎢ ⎢ ⎡ ψ_{4} ψ_{3} ψ_{2} ψ_{1} ⎦ ⎥ ⎥ ⎥ ⎤ + \partial_{y} ⎣ ⎢ ⎢ ⎢ ⎡ - ψ_{4} ψ_{3} - ψ_{2} ψ_{1} ⎦ ⎥ ⎥ ⎥ ⎤ - i \partial_{z} ⎣ ⎢ ⎢ ⎢ ⎡ ψ_{3} - ψ_{4} ψ_{1} - ψ_{2} ⎦ ⎥ ⎥ ⎥ ⎤ + m ⎣ ⎢ ⎢ ⎢ ⎡ ψ_{1} ψ_{2} - ψ_{3} - ψ_{4} ⎦ ⎥ ⎥ ⎥ ⎤

(1)

Equation 1.

Expanded Dirac equation in Planck units

Then as done at physics.stackexchange.com/questions/32422/qm-without-complex-numbers/557600#557600 from why are complex numbers used in the Schrodinger equation?, we could further split those equations up into a system of 8 equations on 8 real-valued functions.

Video 1.

Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)

Source.

Video 2.

PHYS 485 Lecture 14: The Dirac Equation by Roger Moore (2016)

Source.

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Dirac equation solution for the hydrogen atom  Updated 2024-12-15  +Created 1970-01-01

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Predicts fine structure.

Bibliography:

Video "Quantum Mechanics 12b - Dirac Equation II by ViaScience (2015)" youtu.be/tR6UebCvFqE?t=549

Video 1.

Dirac equation for the electron and hydrogen Hamiltonian by Barton Zwiebach (2019)

Source. Uses perturbation theory to get to the relativistic corrections of fine structure! Part of MIT 8.06 Quantum Physics III, Spring 2018 by Barton Zwiebach

Video 2.

How To Solve The Dirac Equation For The Hydrogen Atom | Relativistic Quantum Mechanics by Dietterich Labs (2018)

Source.

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Gross hydrogen emission spectrum  Updated 2024-12-15  +Created 1970-01-01

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One reasonable and memorable approximation excluding any fine structure is:

E_{n} = - \frac{1 3 . 6 e V}{n ^{2}}

(1)

Equation 1.

Hydrogen spectral series mnemonic

see for example example: hydrogen 1-2 spectral line.

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Hydrogen spectral series  Updated 2024-12-15  +Created 1970-01-01

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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?

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Hyperfine structure  Updated 2024-12-15  +Created 1970-01-01

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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:

hydrogen line useful in astronomy, and also the simplest possible case between 1s
caesium standard, which is used to define the second in the International System of Units since 1967.

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NIST Atomic Spectra Database  Updated 2024-12-15  +Created 1970-01-01

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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 $2$ 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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Schrödinger equation  Updated 2024-12-15  +Created 1970-01-01

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The partial differential equation of non-relativistic quantum mechanics.

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:

i ℏ \frac{\partial ψ ( x , t )}{\partial t} = \hat{H} [ψ (x, t)]

(1)

Equation 1.

Schrodinger equation

where:

$ℏ$ is the reduced Planck constant
$ψ$ is the wave function
$t$ is the time
$\hat{H}$ 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.:

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

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.

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

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