Ciro Santilli @cirosantilli 35

 Incoming links: Zeeman effect

1902 Nobel Prize in Physics  Updated 2024-12-23  +Created 1970-01-01

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Pieter Zeeman for the Zeeman effect.

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Angular momentum operator  Updated 2024-12-23  +Created 1970-01-01

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Basically the operators are just analogous to the classical ones e.g. the classical:

L_{z} = x p_{y} - y p_{x}

(1)

becomes:

\hat{L}_{z} = - i ℏ (x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x})

(2)

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

\hat{L}^{2} = \hat{L}_{x} + \hat{L}_{y} + \hat{L}_{z}

(3)

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 1. "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 experiment. Likely Zeeman effect.

Video 1.

Quantum Mechanics 7a - Angular Momentum I by ViaScience (2013)

Source.

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B3 Oxford physics course  Updated 2024-12-23  +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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Schrödinger equation  Updated 2024-12-23  +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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Spectral line  Updated 2024-12-23  +Created 1970-01-01

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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

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