Stark Effect Lab (2D)

Linear Stark effect in hydrogen

Hydrogen's levels with the same n are degenerate, so an electric field F mixes them at first order. In parabolic coordinates the energies are E = −13.6 eV/n2 + (3/2) n (n1 − n2) e a0 F with n1 + n2 + |m| + 1 = n. The unit e a0 F is 0.0529 meV per MV/m, so for n = 2 the shifts are 0, 0 and ±3 e a₀ F. This page also diagonalises the matrix of z in the n manifold numerically, as an independent check.

The Lamb shift and the weak-field limit

In real hydrogen 2s lies 1058 MHz (4.37 µeV) above 2p_1/2. The two-level matrix [[−Δ/2, W],[W, Δ/2]] with W = 3 e a0 F gives E = ±√(Δ2/4 + W2): quadratic in F for W << Δ/2, linear above the crossover field (about 14 kV/m here).

Quadratic Stark effect

A non-degenerate level has no first-order shift; second order gives ΔE = −½ α F2 with the static polarizability α. For the hydrogen ground state α is exactly 9/2 atomic units; sodium 3s is about 163 and caesium 6s about 401 a.u., which is why alkali atoms are far more sensitive to fields.

Honest simplifications

The fan diagram uses only the electrostatic coupling inside one n manifold (no mixing between different n, no fine structure beyond the 2s-2p Lamb picture, no field ionisation and no Inglis-Teller overlap). The 3D page uses an abstract field unit and a made-up coefficient, so its sublevel energies are schematic; here the energies are in real units. Spectral line intensities and polarisation selection rules are not modelled.

• Start at n = 2: two sublevels move up and down linearly while two stay put; the coloured dots sit exactly on the fan lines.
• Pick Hydrogen n = 4: the fan has 16 sublevels but only 7 distinct energies, because states share the same n1 - n2.
• Pick Very weak field: on the first graph the shift bends from quadratic into linear around the crossover field.
• Compare Sodium and Hydrogen ground state: the same field shifts sodium about 36 times more.