Quantum mechanics describes the Zeeman effect as the splitting
of energy leves. It is caused by the so called "m" quantum number.
This effectively quantises the orientation of the electrons orbit.
m can take values from -n to n where n is the principle quantum
number. for example if n = 1 then m = -1,0,1. The n=1, m = -1 and
the n=1, m=0 quantum staes have slightly different energies and
this leads to the splitting of the energy levels as observed by the
Zeeman effect