The Stark
The experimenters had been waiting for this for years.
- Sommerfeld and Debye, working almost simultaneously, both predicted it: put a hydrogen atom in an electric field, and its spectral lines will split. Not broaden, not blur—split. Cleanly. Symmetrically. Like a fork in the road that the atom is forced to choose.
And then Zeeman and Stark themselves verified it. The line didn't just shift. It divided. Into three components.
The story begins with a contradiction. The ground state of hydrogen ($n=1$, $l=0$, the 1s orbital) has a definite parity—it's even. Flip the coordinates: $\vec{r} \to -\vec{r}$, and the wavefunction stays the same. The perturbation from an electric field is $\hat{H}' = e\mathcal{E}z$, which is odd under parity. The matrix element $\langle 1s | z | 1s \rangle$ is an integral of an odd function over all space. It's zero. By symmetry, by parity, the first-order Stark shift of the ground state vanishes. The ground state is stubborn.
But the excited states are a different matter. Take $n=2$. Hydrogen has a famous degeneracy—states with different $l$ but the same $n$ share the same energy. The 2s ($l=0$) and 2p ($l=1$, three orientations) are degenerate (in the non-relativistic picture, ignoring fine structure). That fourfold degeneracy is the key.
The electric field breaks the spherical symmetry. It picks out a direction—the $z$ axis. Suddenly, $l$ is no longer a good quantum number. What remains good is the projection of angular momentum along the field direction, $m_\ell$. The $m_\ell = 0$ states can mix (they have the right symmetry), and the field pulls the 2s and 2p$_z$ together.
Diagonalize the perturbation in the degenerate $n=2$ subspace, and you find: three of the states remain at the original energy (first-order shift = 0), and two split apart by $\pm 3e\mathcal{E}a_0$. One shifts up. One shifts down. The central line stays. Three components.
That's what the experimenters saw. A spectral line—say, the Balmer-alpha transition—that was originally a single sharp line, now fanning out into three. The outer two shifted symmetrically by an amount proportional to the field strength. Linear Stark effect.
But here's what made the hydrogen atom special: the linear Stark effect required degeneracy. Without it, you'd only get the quadratic effect (second-order perturbation), where the shift is proportional to $\mathcal{E}^2$ rather than $\mathcal{E}$. All atoms without the accidental $n$-degeneracy of hydrogen show only the quadratic Stark effect. The shift is tiny—proportional to the field squared—and it's really the polarizability of the atom in disguise.
The quadratic effect works like this: the electric field induces a dipole moment. The field distorts the electron cloud, slightly shifting the center of negative charge away from the nucleus. That induced dipole then interacts with the field, lowering the energy. $E^{(2)} = -\frac{1}{2}\alpha \mathcal{E}^2$, where $\alpha$ is the polarizability. The ground-state hydrogen atom has $\alpha = \frac{9}{2}a_0^3$—a clean, computable number. The atom gets squished. The energy drops. The spectral line shifts by a small, field-squared amount.
For hydrogen excited states, both effects coexist. The degenerate states give the linear splitting. The non-degenerate partners (states with different $n$ that mix at second order) give a smaller quadratic contribution. The full Stark pattern is a messy superposition—but the linear part dominates at high field.
What made the Stark effect historically important was that it provided direct evidence for the quantum numbers of the hydrogen atom. The pattern of splitting encoded the angular momentum structure. It confirmed that $l$ and $m_\ell$ were real—physical, measurable quantities, not just mathematical conveniences. The spectral lines spoke.
And the degenerate perturbation theory you needed to calculate it—diagonalizing a matrix in the degenerate subspace—became one of the first dramatic applications of that technique. It was the proof that perturbation theory could handle the real, complicated spectrum of an atom with real degeneracies and real symmetries.
Today, the Stark effect is a practical tool. Stark spectroscopy—applying fields and watching lines split—gives you information about energy levels, selection rules, and transition moments. It's used in plasma diagnostics, in measuring electric fields in stars, in controlling the energy levels of Rydberg atoms for quantum information experiments.
A single electric field. A hydrogen atom. Lines splitting into three.
Sometimes the universe is that simple.