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Last modified: November 20, 2021 ✓

Zeeman Effect
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The magnetic dipole moment is given by:





μ = IA

The torque on the dipole moment is given by:

τ = μ x B 

The torque tends to line up the magnetic 
dipole moment with the magnetic field 
because this represents the lowest energy 
configuration.

The potential energy associated with the 
dipole moment is:

U(θ) = -μ.B 

Therefore, the change in energy between 
the up and down state is:

ΔE = 2μB

The magnetic dipole moment for the 
electron is:

μe = -eL/2m

Therefore the energy becomes:

E = eBL/2m

For the orbital angular momentum in 
the z-direction, 

L = ml.  Thus we can write:

E = ehBml/2m

  = μBmlB 

Where μB is the Bohr Magneton = eh/2m

This gives a displacement in the energy 
levels such that:

ΔE = mlμBB 

This splitting of the levels is referred 
to as the ZEEMAN EFFECT.

Similarly, for the spin angular momentum,
MS = ±1/2h.  Thus, we can write:

E = ehBmS/2m

  = μBmSB 

In general, the Zeeman effect is the result 
of both the orbital and spin angular momenta.  
Thus, we can write:

ΔE = (e/2m)(L + gS).B = gLμBmJB

Where g is the spin g-factor ~ 2 and gL 
is the LANDE g-factor

The Lande g-factor is a geometric factor 
that compensates for the fact that the S 
and L vectors are both precessing around 
the magnetic field in generally different 
directions.

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