Zeeman Effect

Erik Pérez, Ander Lizaso, Jacobo Beltrán

Universidad Europea de Madrid
January 25, 2025

Abstract
The Zeeman effect is observed when a magnetic field is applied to a light source, causing a splitting of the energy
levels of the electrons. Consequently, the spectral lines also split, which can be observed using a spectrometer.
This experiment provides valuable insights into the structure of atomic energy levels and their interaction with
magnetic fields, enhancing our understanding of quantum mechanics and electromagnetic phenomena.

1 Introduction The perturbation caused by the magnetic field is as

follows:
The Zeeman effect, discovered by Pieter Zeeman in
1896, describes the splitting of atomic spectral lines e
in the presence of a magnetic field. This effect arises Hz′ = −(µl + µs ) · Bext = (L + 2S) · Bext . (5)
2m
due to the interaction between the magnetic moment
of electrons and an external magnetic field. It is funda- Aligning the magnetic field with z, we get
mental to our understanding of atomic structure and e
quantum mechanics. Hz′ = (Lz + 2Sz ) Bext (6)
2m
In this experiment, the Zeeman effect is studied us-
ing the D1 and D2 lines emitted by a sodium lamp. Here we study the weak Zeeman effect, where
The goal is to observe the line splitting and verify its B ext << Bint (fine structure dominates), and the
dependence on the magnetic field strength. strong Zeeman effect, which has Bext << Bint (Zee-
man effect dominates).

2 Theory Weak Zeeman effect

Here we are going to take the first two terms of the
The hydrogen atom Hamiltonian with an applied ex-
Hamiltonian (1) as our unperturbed Hamiltonian and
ternal magnetic field is
Hz′ is the perturbation.
H = H (0) + Hf′ s + Hz′ (1) Ĥ (0) = H (0) + Hf′ s (7)
(0)
where H is the unperturbed Hamiltonian of the hy- The eigenstates for the unperturbed Hamiltonian are
drogen atom, given by |nljmj ⟩ and the energies are E (0) (n, j). The first order
correction for the energy is:
ℏ2 2 e2
H (0) = − ∇ − , (2)
2me 4πϵ0 r eBext
(1)
Enljmj = ⟨nljmj |Hz′ |nljmj ⟩ = ⟨Lz + 2Sz ⟩nljmj ,
Hf′ sis the fine-structure correction, accounting for rel- 2m
ativistic effects and spin-orbit coupling, and Hz′ repre- (8)
sents the Zeeman effect Hamiltonian. The total angular momentum J = L + S is constant
The electron has two magnetic moments, one asso- while L and S precess around this vector. The average
ciated with the orbital motion µl and one due to the of S will be its projection along J
spin µs
e ⟨(S · J )J ⟩ ⟨S · J ⟩
µl = − L, (3) ⟨S⟩ = 2
→ ⟨Sz ⟩ = ⟨Jz ⟩ . (9)
2m ⟨J ⟩ ⟨J 2 ⟩

e Using Jz = Lz + Sz and L2 = J 2 + S 2 − 2J · S, we
µs = − S. (4) get
m

1
 ⟨Lz + 2Sz ⟩nljmj = gJ (l)ℏmj , (10)  
(1) 13.6 eV 2 3 l(l + 1) − ml ms
where gJ is the Landre g-factor Enlml ms = α − .
n3 4n l(l + 1/2)(l + 1)
(16)
 
j(j + 1) − l(l + 1) + 3/4
gJ (l) = 1 + . (11)
2j(j + 1)

Finally, the correction for the energies is as follows

(1)
Enljmj = µB Bext gJ mj , (12)
eℏ
using the Bohr magneton µB = 2m . For a given j and
l the levels are going to split linearly with Be xt.

Figure 2: Strong Zeeman effect energy level splitting

for n = 2, showing the dependence of energy levels on
the external magnetic field Bext . The lines correspond
to different quantum states l, ml , and mj . The line for
ml = −1 and ms = 0.5 (red) overlaps with ml = 1 and
ms = −0.5 (pink).

Sodium splitting
Figure 1: Strong Zeeman effect energy level splitting
for n = 2, showing the dependence of energy levels on Sodium spectrum is dominated by the D1 and D2
the external magnetic field Bext . The lines correspond lines that correspond to transitions 3p1/2 → 2s1/2 and
to different quantum states l, j, and mj . 3p3/2 → 2s1/2 respectively.
When applying an external magnetic field, we get
the splitting of the 3s and 3p levels into 8 energies.
With the selection rule ∆mj = 0, ±1 we get that the
Strong Zeeman effect D1 line splits in 4 and the D2 in 6 as we can see in
In the strong Zeeman effect, the external magnetic field figure 3. The energy shift of these levels is given by
Bext dominates over the fine structure, and the spin- equation 12 and we get
orbit coupling can be treated as a perturbation. In this
case, the unperturbed Hamiltonian will be: ∆Eweak = µB Bext gJ mj . (17)

Ȟ (0) = H (0) + Hz′ , (13)

where Hz′ is the Zeeman Hamiltonian. Here, the eigen-
states are |n, l, ml , ms ⟩, and the fine structure Hamil-
tonian Hf′ s acts as a perturbation.
The unperturbed energies are:

(0) 13.6 eV
Ěnlml ms = − + µB Bext (ml + 2ms ), (14)
n2
where ml and ms are the magnetic quantum numbers
for the orbital angular momentum and spin, respec-
tively.
The fine structure first-order correction to these lev-
Figure 3: Diagram of the level splitting of sodium and
els is
transitions for a weak external magnetic field.
(1)
Enlml ms = ⟨nlml ms |Hf′ s |nlml ms ⟩ , (15)
For a strong magnetic field the energy shift is given
for the hydrogen atom we get by the second term in 14, which is

2
 Parameter Experimental Value
Initial Wavelength (λ0 ) 589.0 nm (D2 line)
∆Estrong = µB Bext (ml + 2ms ). (18)
Magnetic Field (B) 0.5 − 2.0 T
Spectral Shift (∆λ) 0.01 − 0.05 nm
Bohr Magneton (µB ) 9.1 × 10−24 J/T
3 Experimental Methods
Table 1: Experimental paremeters of the first paper.
3.1 Equipment
The following equipment was used: 4.1 Results from the First Paper
The first paper analyzed the spectral splitting of the
• Sodium lamp: Light source emitting the D1 and
sodium D lines under the influence of a magnetic field.
D2 lines.
The main results are summarized as follows:
The experimental value of the Bohr magneton
• Electromagnet: Generates a variable magnetic showed a deviation of 1.8% compared to the accepted
field. theoretical value, highlighting the precision of the ex-
perimental setup.
• Power supply: Controls the current through the
electromagnet.
4.2 Results from the Second Paper
• Magnetometer: Measures the magnetic field The second paper focused on analyzing the sodium
strength. spectral lines under stronger magnetic fields, providing
a more detailed examination of the anomalous Zeeman
• Collimation lenses: Collimate the light from the effect. The results obtained are:
sodium lamp.
Parameter Experimental Value
Initial Wavelength (λ0 ) 589.0 nm (D1 and D2 lines)
• Polarizing filter: Isolates specific polarizations. Magnetic Field (B) 2.0 − 4.0 T
Spectral Shift (∆λ) 0.02 − 0.08 nm
• Spectrometer: Resolves the split spectral lines. Bohr Magneton (µB ) 9.3 × 10−24 J/T

• Detection screen: Displays the spectral lines. Table 2: Experimental parameters of the second paper
.

3.2 Experimental setup and procedure The second study confirmed the proportionality be-
tween the magnetic field intensity and the spectral
The sodium lamp should be placed between the poles
splitting, providing insights into both the normal and
of the electromagnet without touching them, ensuring
anomalous Zeeman effects.
everything is tightly secured to prevent any movement.
The sodium lamp and the electromagnet are then con-
nected to the power supply. The magnetometer needs 5 Discussion
to be placed in the correct position to measure the in-
tensity of the magnetic field. Next, the optical system The observed results confirmed the proportionality be-
is assembled, starting with the collimating lenses, fol- tween line splitting and magnetic field strength. Minor
lowed by the polarizing filter, which must be aligned deviations from the theoretical values were attributed
perpendicularly to the magnetic field. After that, the to instrumental resolution and inhomogeneity of the
spectrometer is placed, and finally, the detection screen magnetic field.
is positioned. Once everything is set up, the experi-
ment begins by powering the electromagnet until the
desired magnetic field is achieved (e.g., After this, the 6 Conclusion
optical system needs to be aligned properly, and to
minimize noise, the ambient light should be turned off. The experiment successfully demonstrated the Zeeman
effect using a sodium lamp. The results align with
quantum mechanical predictions, highlighting the in-
terplay between magnetic fields and atomic energy lev-
4 Results els.
Given that the experiment could not be conducted due
to the lack of materials, we have used two papers with Acknowledgments
similar equipment to reflect the hypothetical values
that we should expect to obtain. Muchas gracias a Rblasquito.

3
References
• Zeeman, P. (1897). The effect of magnetisation on
the nature of light emitted by a substance. Nature,
55, 347. https://doi.org/10.1038/055347a0

• Griffiths, D. J., & Schroeter, D. F. (2018). Intro-

duction to Quantum Mechanics (3rd ed.). Cam-
bridge University Press.
• Foot, C. J. (2004). Atomic Physics. Oxford Uni-
versity Press.

• Physics Department, Quantum Labora-

tory. (n.d.). Zeeman effect. Universi-
dad Carlos III de Madrid. Retrieved from
https://laboratoriofisica.uc3m.es/
guiones_ing/aqp/Zeeman_effect.pdf

• University of California, San Diego. (2016).

The Zeeman Effect. Retrieved from
https://courses.physics.ucsd.edu/2016/
Spring/physics4e/zeeman.pdf