The Zeeman Effect (Intro)

Place a hydrogen atom in a magnetic field and its spectral lines split into several components. This is the Zeeman effect, discovered in 1896, and it is direct experimental proof that the magnetic quantum number $m$ is real and that angular momentum is quantized in space.

The magnetic moment of an orbiting electron

An electron with orbital angular momentum $\mathbf{L}$ has a magnetic dipole moment

The constant $\mu_B$ is the Bohr magneton, the natural unit of atomic magnetism.

The interaction energy

A magnetic moment in a field $\mathbf{B} = B\hat{z}$ has energy $H' = -\boldsymbol{\mu}\cdot\mathbf{B}$. For the orbital moment,

Because the unperturbed hydrogen states are already eigenstates of $L_z$ with eigenvalue $\hbar m$, this perturbation is diagonal: each state simply shifts by

Splitting of a level

A level with orbital quantum number $\ell$ contains the $2\ell+1$ values $m = -\ell, \ldots, +\ell$. In zero field these are degenerate; switching on $B$ fans them into $2\ell+1$ equally spaced sublevels separated by

For example, a $p$ level ($\ell = 1$) splits into three lines with $m = -1, 0, +1$. This evenly spaced triplet is the normal Zeeman effect. The selection rule $\Delta m = 0, \pm 1$ then predicts the observed pattern of shifted spectral lines.

Orders of magnitude

In a strong laboratory field of $B = 1\,\text{T}$, the splitting is $\mu_B B \approx 5.8 \times 10^{-5}\,\text{eV}$ — about a part in $10^5$ of the $-13.6\,\text{eV}$ ground state. Tiny, but easily resolved spectroscopically, which is why the Zeeman effect is a standard tool for measuring magnetic fields in plasmas and on the surfaces of stars.

The takeaway

A uniform field $B\hat{z}$ shifts each hydrogen sublevel by $\Delta E = \mu_B B m$, splitting an $\ell$-level into $2\ell+1$ equally spaced lines separated by $\mu_B B$, with the Bohr magneton $\mu_B \approx 5.788 \times 10^{-5}\,\text{eV/T}$.

Try it

This is a numerical exercise — return a number. Using $\mu_B = 5.788 \times 10^{-5}\,\text{eV/T}$ and $B = 2\,\text{T}$, compute the energy spacing (in eV) between adjacent $m$ sublevels in the normal Zeeman effect.

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