The Zeeman effect is the splitting of a spectral line by a magnetic field. That is, if an atomic spectral line of 400 nm was considered under normal conditions, in a strong magnetic field, because of the Zeeman effect, the spectral line would be split to yield a more energetic line and a less energetic line, in addition to the original line at 400 nm.
The reason for the Zeeman effect is that in a magnetic field, the angular momentum quantum state can undergo a displacement from degeneracy. For example, the p orbital has three possible angular momentum quantum states that are degenerate (of the same energy) under normal circumstances. However, each angular momentum quantum state has a magnetic dipole moment associated with it, so the effect of a magnetic field is to separate the three states into three different energy levels. One state elevates in energy, one lowers in energy, and one remains at the same energy. The separation of these quantum states into three different energy levels results in 3 different excitation states with slightly different energies that give rise to three spectral lines of slightly different energy (one of the same energy as the original spectral line, one more energetic, and one less energetic) upon relaxation of the atom. This is the simplist case of the Zeeman effect, known as the Normal Zeeman effect.
Answered by:
Jay Foley, Undergraduate Chemistry Student, Georgia Tech

In a magnetic field the energy of a particular atomic stat depends on the value of m ,magnetic quantum number,. A state of total quantum number n breaks up into several substates when the atom is in the magnetic field ,and their energies are slightly more or slightly less than the energy of state in the absence of magnetic field. This phenomenon leads to "splitting" of individual spectral lines when atoms radiate in a magnetic field. The spacing of the lines depends on the magnitude of the field.
The splitting of the spectral lines by magnetic field is called ZEEMAN EFFECT. The Zeeman effect is a vivid confirmation of space quantisation.
The normal Zeeman effect consists of splitting of a spectral line of frequency 'f' into three components whose frequencies are:
f(1) = f - (22/7)eB/4m
f(2) = f
f(3) = f + (22/7)eB/4m

Answered by:
Ashok kumar Sharma, Technology Student, NIT JSR, INDIA

The Zeeman effect is the splitting of the spectral lines of an atom in the presence of a strong magnetic field. The effect is due to the distortion of the electron orbitals because of the magnetic field.
The (normal) Zeeman effect can be understood classically, as Lorentz predicted. Zeeman discovered the effect, but under closer investigation it did not agree with Lorentz. These differences were explained by the quantum mechanics effects of spin. This is the anomalous Zeeman effect.
In fact, it was the anomalous Zeeman effect that led to the discovery of spin.
Any book on quantum mechanics will deal with the Zeeman effect. The usual method is to use perturbation theory, the details depend on the strength of the magnetic field. Consider the hydrogen atom. The full Hamiltonian H is split up into 3 pieces.
H = Hcoulomb + Hrelativistic + Hzeeman
Here Hcoulomb is the Coulomb term, Hrelativistic are the relativistic correction terms and Hzeeman is the Zeeman term produced by the magnetic field.
To first-order the relativistic terms led to the fine-structure energy shift.
If the magnetic field is weak compared to the relativistic corrections then the Zeeman term can be considered a perturbation of the relativistic terms.
If the magnetic field is strong then we can diagonalize the coulomb and Zeeman terms and then consider the relativistic correction as a perturbation. This is the Paschen-Back Effect.
For arbitrary magnetic fields degenerate perturbation theory is needed.
For arbitrary atoms it becomes difficult, but the same ideas apply.
A very similar effect is the Stark effect in which the atom is placed inside a strong electric field. Again, perturbation theory is the usual approach.
Answered by:
Andrew James Bruce, Grad student, UK

'A theory with mathematical beauty is more likely to be correct than an ugly one that fits some experimental data. God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe.'