Stark Effect

The Stark effect is now a tool wherever a field needs measuring or a level needs moving. Rydberg-atom electrometry inverts this article's scalings: because Stark shifts and ionization thresholds blow up so fast with $n$, a vapor cell of Rydberg atoms is a calibrated, SI-traceable antenna reading radio-frequency and microwave amplitudes off Stark-shifted lines, with the $1/(16n^4)$ ramp as state-selective detector. In precision timekeeping the enemy version dominates: blackbody radiation at 300 K is a bath of oscillating fields whose time-averaged Stark shift — the quadratic branch — is among the largest systematics optical clocks must subtract at the $10^{-18}$ level. Two further habitats: in plasmas, fluctuating ion microfields Stark-broaden emission lines, making line widths a remote thermometer-and-densitometer from tokamaks to white-dwarf atmospheres; in semiconductor quantum wells the quantum-confined Stark effect — the exciton red shift under bias — drives the electro-absorption modulators that put data onto optical fiber. The AC-Stark (light) shift belongs to the time-dependent chapter, the giant response of polar molecules to the molecular and solid-state tracks.

Two atoms answer the same field two ways, and the difference is whether they must build a dipole or already own one. The generic atom — sodium's ground state, helium, hydrogen $1s$ — is a definite-parity charge cloud centered on the nucleus. Switch on the field and the cloud shifts against the nucleus in proportion to it: an induced dipole $\boldsymbol{\mu}_{\text{ind}} = \alpha\boldsymbol{\mathcal{E}}$. The energy gain is $-\tfrac{1}{2}\alpha\mathcal{E}^2$ — quadratic, the $\tfrac{1}{2}$ because half the would-be gain is spent distorting the cloud, like the $\tfrac{1}{2}CV^2$ stored while charging a capacitor. Hydrogen's excited shells are the exception. The $2s$ and $2p_0$ states sit at the same unperturbed energy — the accident of Accidental Degeneracy and the SO(4) Symmetry — and have opposite parity. The field distorts nothing: it picks the two ready-made hybrids, one lopsided upfield, one downfield, which behave like permanent dipoles of fixed size. Their energies split linearly in $\mathcal{E}$, no factor-$\tfrac{1}{2}$ penalty, no smallness.

Symmetry referees the affair. The field singles out one axis. Rotations about it survive, so $\hat{L}_z$ still commutes with the full Hamiltonian and $m$ stays good — the conservation logic of Degeneracy and the Role of the Magnetic Quantum Number in a broken world. Rotations that tilt the axis are gone, so $\ell$ dies as a label. And because the field couples to position, not spin, the electron's magnetic moment is a spectator — the sharpest contrast with the sibling Zeeman Effect, where the doubled spin moment causes all the drama.

Before any perturbation theory, the operator. A uniform electrostatic field descends from the scalar potential $\phi(\mathbf{r}) = -\boldsymbol{\mathcal{E}}\cdot\mathbf{r}$, and a charge $q$ in it costs energy $q\phi$. The electron carries $q = -e$ ($e > 0$), so a field along $\hat{\mathbf{z}}$ adds one term to the atomic Hamiltonian:

where: $\hat{\boldsymbol{\mu}} = -e\hat{\mathbf{r}}$ is the electric dipole operator, $\mathcal{E}$ the field magnitude, $\hat{z} = \hat{r}\cos\theta$ the position component along the field. The operator simply prices displacement: cloud pushed upfield costs energy, downfield saves it. The nucleus, pinned at the origin, contributes only a droppable constant. One structural fact does enormous work later: $\cos\theta$ is, up to normalization, the single spherical harmonic $Y_1^0$ from Associated Legendre Functions,

where: no $\varphi$-dependence means $\hat{z}$ commutes with $\hat{L}_z$, and the single unit of angular structure means $\hat{z}$ can shuffle $\ell$ by at most one. Equation (2) is the selection theorem in embryo: a parity-odd, $m$-blind, one-notch-of-$\ell$ perturbation. Watch the sign convention: with electron charge $-e$ ($e>0$), $U = q\phi = (-e)(-\mathcal{E}z) = +e\mathcal{E}z$, so positive $\langle z\rangle$ raises the energy; the identity $Y_1^0 = \sqrt{3/4\pi}\cos\theta$ and the recovery $-\partial_z\phi = \mathcal{E}$ are checked symbolically in this article's verify script. Notice what $\hat{H}'$ does not contain: spin. A static electric field speaks to where the electron is, not to its magnetic moment, so the machinery of Fine Structure of Hydrogen enters this chapter only as an energy landscape — deciding what counts as "degenerate" — never as a new interaction.

Naively, the first-order machinery of Non-Degenerate Time-Independent Perturbation Theory hands us a shift proportional to the state's average displacement:

where: $\langle\hat{z}\rangle$ is the state's permanent dipole moment in units of $-e$. For it to be nonzero the cloud would have to be lopsided before the field arrives. Definite-parity states are never lopsided, and central-potential eigenstates have parity $(-1)^\ell$ — the three-dimensional descendant of Parity Symmetry of One-Dimensional Eigenstates. So equation (3) is zero for every non-degenerate level; the theorem below sharpens this into the full traffic law.

The theorem's two faces run the chapter: diagonals vanish, so a non-degenerate level starts responding at second order; allowed off-diagonals survive, so if two $\Delta\ell = \pm1$, $\Delta m = 0$ partners are degenerate, the perturbation acts inside the manifold at full first-order strength. Which face you meet is the whole phenomenology.

The quadratic branch. For a non-degenerate state the leading response is the sum-over-states machinery of Second-Order Energy Correction with $\hat{H}' = e\mathcal{E}\hat{z}$:

where: all field dependence sits in the explicit $\mathcal{E}^2$, and the atom-specific sum defines the static polarizability $\alpha$. For a ground state every denominator is negative, so $\alpha > 0$ — the sign theorem of Second-Order Energy Correction: ground states are pushed down. Evaluating $\alpha$, including the trick that tames the sum for hydrogen, is the property of Quadratic Stark Effect and Polarizability; here we need only the structure — quadratic, negative for ground states, entirely parity's doing.

The linear branch. Hydrogen's shell $n$ holds all $\ell = 0, \dots, n-1$ at one energy, so opposite-parity partners are degenerate and the theorem's second face takes over. Non-degenerate formulas divide by zero; the legal tool is the W-matrix of Degenerate Perturbation Theory: restrict $\hat{H}'$ to the shell and diagonalize

where: $a, b$ run over the $n^2$ shell states and the eigenvalues are the first-order shifts. The selection theorem is now a gift: it zeroes almost every entry, chopping $W$ into blocks labeled by $m$ — bookkeeping certified by Good Quantum Numbers and Symmetry-Adapted Bases, since $\hat{L}_z$ commutes with both $\hat{H}_0$ and $\hat{H}'$. For $n = 2$ a single $2\times2$ block couples $2s$ to $2p_0$ while $2p_{\pm1}$ sit untouched; its eigenvalues split linearly, $\pm|W_{12}|$ by the two-level closed form of Degenerate Perturbation Theory. The matrix element's value, the level diagram, and the manufactured dipole are computed — radial integral, sign and all — in Linear Stark Effect in Hydrogen.

The umbrella formula. The linear branch has a closed form for every shell, in the coordinates the field prefers: parabolic labels $(n_1, n_2, m)$ with $n_1 + n_2 + |m| + 1 = n$ replace $(\ell, m)$, and the first-order energies are

where: $n_1 - n_2$ is the integer dipole asymmetry of the state along the field; $m$ does not enter at first order. Equation (6) is quoted as the umbrella result — the parabolic separation producing it, and the $n = 2$ cross-check, belong to Linear Stark Effect in Hydrogen. Two structural facts are verified here: the label census gives exactly $n^2$ states per shell ($n \leq 6$ checked), and $|n_1 - n_2|_{\max} = n - 1$, so the fan's width grows like $n^2\mathcal{E}$ — the first hint of how field-sensitive the $n = 50$ Rydberg atoms of The Bohr Energy Spectrum and Rydberg Constant are. Which branch is legal also depends on the competition. The pure-Coulomb treatment assumes the Stark coupling dwarfs the fine-structure splittings separating $2s_{1/2}$, $2p_{1/2}$, $2p_{3/2}$; in the opposite limit fine structure owns the manifold, its surviving degeneracies carry equal parity, and even hydrogen turns quadratic. The crossover is computed in the worked examples.

Perturbation theory of any order still lies about one thing. Add the field's potential to the Coulomb well and look along the downfield axis:

where: the second term descends without bound as $z \to +\infty$ (downfield taken as $+z$ for the electron), so the global spectrum is a continuum — somewhere is always energetically better than the atom. What the field cannot always provide is a classical path out: a barrier stands in the way, enormous at laboratory fields. Every Stark "level" is really a resonance, astronomically long-lived below threshold, collapsing as the barrier top descends to meet it.

In SI units the atomic field $e/(4\pi\varepsilon_0 a_0^2) = 5.14\times10^{11}$ V/m — from The Bohr Radius and Characteristic Atomic Scales — sets the exchange rate: ground-state hydrogen holds to $3.2\times10^{10}$ V/m, hopeless in a laboratory, while an $n = 50$ Rydberg atom lets go at $5.1\times10^{3}$ V/m — 51 V across a centimeter. That collapse is the $n^4$ law at work.

Example 1 — the $n = 2$ shell: reading the structure before computing anything. A field $\mathcal{E}$ is applied to hydrogen's first excited shell. Which states move, and when is the pure-Coulomb treatment legal?

Step 1. List the shell: $|200\rangle$, $|210\rangle$, $|21{,}{+}1\rangle$, $|21{,}{-}1\rangle$ — four states, one energy $E_2 = -3.4$ eV from The Bohr Energy Spectrum and Rydberg Constant. Degenerate manifold ⇒ the fork sends us to the W-matrix, equation (5).

Step 2. Apply the selection theorem entry by entry. Diagonals: all vanish (parity). Off-diagonals: only $\Delta\ell = \pm1$ and $\Delta m = 0$ pairs survive — the single pair $\{|200\rangle, |210\rangle\}$.

Step 3. Read off the structure: $W$ is block-diagonal — one $2\times2$ block $\begin{pmatrix} 0 & W_{12}\\ W_{12}^* & 0\end{pmatrix}$ for the $m = 0$ pair, two zero blocks for $m = \pm1$. By the two-level closed form of Degenerate Perturbation Theory, the shell splits into three tiers: one level up, one down, an untouched $m = \pm1$ doublet — the size $|W_{12}|$, its radial integral and sign computed in Linear Stark Effect in Hydrogen.

Step 4. Legality: we ignored fine structure, so we need $e\mathcal{E}a_0 \gg \Delta E_{fs}(n{=}2) = 4.53\times10^{-5}$ eV (the $2p_{3/2}$–$2p_{1/2}$ gap, quoted from Fine Structure of Hydrogen). Setting them equal with $a_0 = 5.29\times10^{-11}$ m gives the crossover $\mathcal{E}^* \approx {\bf 8.6\times10^{5}\ V/m}$ (verified) — only $1.7\times10^{-6}$ of the atomic field unit, because fine structure is an $\alpha^2$-suppressed whisper. Well above $\mathcal{E}^*$ the linear fan is honest physics; near or below, the fine-structure basis rules and the response bends quadratic.

Example 2 — field ionization, from hopeless to trivial. What field strips (a) ground-state hydrogen, (b) an $n = 20$ Rydberg atom?

Step 1. The one-sixteenth law in SI: $\mathcal{E}_c = \dfrac{5.14\times10^{11}\ \text{V/m}}{16\,n^4}$, with the atomic field unit from The Bohr Radius and Characteristic Atomic Scales.

Step 2. Ground state: $\mathcal{E}_c = 5.14\times10^{11}/16 = {\bf 3.2\times10^{10}\ V/m}$ (verified) — ground-state atoms field-ionize only in intense pulsed-laser foci or at field-ion-microscope tips.

Step 3. $n = 20$: the threshold falls by $n^4 = 1.6\times10^{5}$, giving $\mathcal{E}_c = {\bf 2.0\times10^{5}\ V/m}$ (verified) — 200 V across a millimeter. This is why every Rydberg experiment ends with a ramped ionization field and an electron detector: the ramp reads out $n$ through the $1/(16n^4)$ law.

• Griffiths and Schroeter, Introduction to Quantum Mechanics, 3rd ed., §7.3 — the Stark problems in this register.
• Sakurai and Napolitano, Modern Quantum Mechanics, 2nd ed., §5.2 — degenerate machinery, Stark doublet as showcase.
• Landau and Lifshitz, Quantum Mechanics, §76–§77 — parabolic coordinates and field ionization.
• Bethe and Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, §51–§55 — the definitive hydrogenic reference.
• Gallagher, Rydberg Atoms — the $1/(16n^4)$ law as an experimental workhorse.

Provenance note: All equations and numbers are standard textbook results. A verifier should: (1) reproduce $\langle200|z|210\rangle=-3a_0$ and the $\pm3ea_0E$ splitting of $n=2$ from Griffiths §7.3 and Sakurai §5.2; (2) confirm the parity argument and the second-order/polarizability formula from any of the listed texts; (3) check $\alpha_{1s}=\tfrac92 a_0^3$ (Gaussian units) via the Dalgarno–Lewis method in Bethe–Salpeter §51 or Landau–Lifshitz §76; (4) verify the parabolic-coordinate separation and the general first-order shift $\tfrac32 n(n_1-n_2)ea_0E$ in Landau–Lifshitz §77. The over-the-barrier/saddle-point field scaling $\sim1/16n^4$ (atomic units) is the standard classical ionization estimate. — The items this article owns (dipole coupling, selection-theorem sweep, explicit-state vanishings, $[\hat{L}_z,\hat{z}]=0$, saddle law with the $1/(9n^4)$ refinement, atomic field unit and all quoted field strengths, parabolic census) are executed in its verify script (41 checks). Edition note: by chapter ownership, the base article's worked derivations of the $n=2$ linear splitting ($\langle200|z|210\rangle = -3a_0$, $\pm3e\mathcal{E}a_0$) and of the parabolic separation behind equation (6) are RELOCATED to Linear Stark Effect in Hydrogen, and its Dalgarno–Lewis polarizability derivation ($\alpha_{1s} = \tfrac{9}{2}a_0^3$) is RELOCATED to Quadratic Stark Effect and Polarizability — the verifier items travel with them. The relocated Dalgarno–Lewis sketch also carries a dimensionally inconsistent prefactor in the base ($-e\mathcal{E}/a_0$ where dimensional analysis requires $-m_e e\mathcal{E}/\hbar^2$); the corrected derivation ships with its owner.

Builds on: Non-Degenerate Time-Independent Perturbation Theory · Second-Order Energy Correction · Degenerate Perturbation Theory · Good Quantum Numbers and Symmetry-Adapted Bases Leads to: Zeeman Effect · Fine Structure of Hydrogen · Lamb Shift