The spin-orbit interaction can be considered as a case of an "internal magnetic field" affecting atomic energy levels. In this section, the influence of external magnetic fields is examined.

When a magnetic dipole is placed in a magnetic field it will experience a torque. Let us analyse this.

Let us consider a “singlet state”. In quantum mechanics, a “singlet state” (term related to the notion of “single spectral line”, is a system where the spin or spin angular momentum is zero and therefore the total angular momentum is equal to the orbital angular momentum. (It is noted that if the spin angular momenta can sum vectorially to zero, the spin is zero.)

As mentioned at https://en.wikipedia.org/wiki/Magnetic_moment, the magnetic (dipole) moment of a magnet is a quantity that determines the torque it will experience in an external magnetic field.

We know how to calculate the torque exerted by the magnetic field on a coil with current I, surface A and n loops. It is described at the link http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html#c2. We will use the same approach.

A magnetic source in general is inherently a dipole source that can be visualized as a coil, or let us say a loop with current I and area A as shown at the picture below from link http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html#c1.

*Figure 2: Image from hyperphysics.phy-astr.gsu.edu *

*http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html#c1.*

The magnetic (dipole) moment can be considered to be a vector μ with a value equal to I * A and a direction perpendicular to the current loop in the right-hand-rule direction. The torque is given by τ = μ * B.

Due to this, a "capacity for doing work" is generated as a result of the magnetic field. The "capacity for doing work" or "potential for doing work" (cf. common expression "this candidate for the job has got potential!") is translated to the notion of "potential energy" as mentioned at http://hyperphysics.phy-astr.gsu.edu/hbase/pegrav.html#pe. In particular, due to the fact that this potential is linked to a magnetic field, this is termed "magnetic potential energy" (cf. http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magpot.html#c1).

The magnetic potential energy U of a magnetic (dipole) moment μ due to the presence of a magnetic field B is given by:

U = - μ * B

The magnetic potential energy is expressed as a scalar product (μ and B are vectors) and it implies that the torque that is exerted tends to line up/to align the magnetic moment with the magnetic field B, so this represents its lowest energy configuration.

The magnetic dipole moment is due to the orbital angular momentum. The relation associating them is:

μorb = - (e/2m) * L

If we consider the magnetic field in the z-direction, given that Lz =mℓ * h, we will have

U = - e/(2m) * Lz * B = mℓ * eh/(2m) * B

This potential energy U will correspond to the change of the energy in the system due to the torque from the magnetic field. Therefore, we will continue with the term ΔΕ.

ΔΕ= mℓ * μΒ * Β where μΒ is the Bohr magnetron

How many values of mℓ (magnetic quantum number) do we have?

For instance for ℓ = 2: mℓ = −2, −1, 0, 1, 2,

In general we have 2ℓ +1 values of mℓ (cf. for ℓ =2 we have 5 values of mℓ above )

This means that 5 different energetic levels will be produced (or can be attributed). In other words, there will be "splitting" in five different levels.

As mentioned at the link http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/sodzee.html#c2 "using the vector model for total angular momentum, the splitting is seen to produce one level for each possible value of the z-component of the total angular momentum J".

It is noted that in this case the splitting will be uniform.

If we have five lines, what are the different energetic jumps that can be made? That is related to the Δmℓ or change of mℓ.

The change of mℓ can be -1, 0 or +1 and there will be only three transition energies:

E0 + ehB/2mℓ,

E0

E0 - ehB/2mℓ

By dividing the change in energy by h, we can find the change of frequency of the emitted spectral line. Therefore, the change of frequency will be:

eB/2mℓ

0

eB/2mℓ

*Figure 1: From the gross structure of the atom to the fine and hyperfine structure. The spin-orbit interaction is linked to the fine
structure and the Zeeman interaction to the hyperfine structure and magnetic sublevels.*

http://demonstrations.wolfram.com/BreitRabiDiagram/

At the reference below at the last page it is demonstrated how we can calculate the magnetic field of the Sun based on the Zeeman effect for sodium.

https://www.atmos-meas-tech.net/8/1863/2015/amt-8-1863-2015.html

"The interaction of the Earth magnetic field with the oxygen dipole leads to a splitting of O_{2} energy states, which polarizes the emission spectra. A special campaign was carried out in
order to measure this effect in the oxygen emission line centered at 53.07 GHz".

Previous satellite studies are cited.

Doctoral Thesis by Richard Larsson

(p.2) "Another application for a Zeeman effect model is in the quantification of magnetic fields. Although in a completely different frequency region and for other molecules
and atoms, the Zeeman effect has been used to determine the magnetic field at the sun and other stars (see, e.g., Berdyugina and Solanki, 2002, and works cited therein)".

(p.iii)" It is possible to utilize the Zeeman effect on molecular oxygen to measure the magnetic field of Mars. The last part of this thesis work suggests a measurement
scheme for a satellite capable of retrieving the horizontal components of the Martian crustal magnetic field."

How we can measure the magnetic field by taking a sample of light!

Video demonstration by ESO featured in Wikipedia article on the Zeeman effect (https://en.wikipedia.org/wiki/Zeeman_effect)

Figure created using their video

Description: Initially, there is one spectral line and as the strength of the magnetic field increases it splits in three components. With further magnetic field strength increase, the distance
between the three spectral lines becomes greater.

20-second demonstration of the Zeeman effect starting at t=120 of video by Synctrotron SOLEIL https://youtu.be/tcpBwgN5hlE?t=120

"By approaching a magnet to the sodium lamp, the characteristic yellow line of sodium is subdivided, proving that the electron experiences an electromagnetic force that modifies its energy levels."

Transcript translation (starting at t=54 - https://youtu.be/tcpBwgN5hlE?t=54): (In 1913) "Bohr's atomic model was describing correctly the hydrogen atom. The electron was rotating in circular orbits corresponding to authorized energy levels. By describing the orbits with positive integers, 1, 2, 3 etc. Niels Bohr was introducing the first quantum number of modern physics. As his model was struggling to describe multi-electron atoms, the German physicist Arnold Sommerfeld improved it in 1916 by providing electrons with two additional degrees of freedom: being able to rotate on elliptical orbits like the planets of the solar system as well as modify their trajectory in the presence of a magnetic field. Sommerfeld was thus adding two numbers: "l" the "orbital quantum number" and "m" the "magnetic quantum number". "

"Magnetic because the electrons behave like a small electrical circuit that is sensitive to external magnetic fields. This is the Zeeman effect, named after the Dutch physicist that discovered it twenty years ago when he studied the sodium spectrum. By approaching a magnet to the sodium lamp, the characteristic yellow line of sodium, is subdivided, proving that the electron experiences an electromagnetic force that modifies its energy levels."

In 1916, the electron thus lives with three quantum numbers."