An external magnetic field shifts atomic energy levels because magnetic moments interact with the field. For orbital angular momentum, the magnetic moment is proportional to $\mathbf L$, so a field along $z$ adds an interaction of the form $$H'=-\boldsymbol\mu\cdot\mathbf B.$$ In the simplest orbital model, $$\Delta E=m_\ell\mu_BB,$$ where $m_\ell$ is the magnetic quantum number and $$\mu_B=\frac{e\hbar}{2m_e}$$ is the Bohr magneton.

A level with fixed $\ell$ but several $m_\ell$ values therefore splits into equally spaced components. For example, an $\ell=1$ level has $m_\ell=-1,0,+1$, producing shifts $$-\mu_BB,\quad0,\quad+\mu_BB.$$ Transitions between the split levels generate multiple nearby spectral lines rather than one unsplit line.

Electron spin contributes an additional magnetic moment and leads to the more general anomalous Zeeman structure. The relevant shift is then expressed in terms of total angular momentum and a Landé $g$ factor.

The Zeeman effect demonstrates three ideas at once: quantum angular-momentum projections are discrete, external fields lift degeneracies, and spectroscopy can measure tiny energy differences. Magnetic resonance and precision atomic measurements build on the same coupling between angular momentum and magnetic fields.