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Last modified: January 26, 2018

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 oribital 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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