Stark–Zeeman Splitting

Choose Stark, orbital Zeeman or collinear E and B. Valid fields update all four n=2 levels immediately. Source and target are chosen in the toolbar; a diagram click selects the target, Shift-click the source, and Enter/Space work too. The comparison arrow connects the selected energies; a dashed arrow has no E1 coupling. Neither an arrow nor equivalent λ proves an emitted photon. Densities toggles a separate explicit density workspace. Both slices use the same logarithmic scale, coordinates −12…12 a₀ and a 5 a₀scale bar. Raster size is bounded at 640 pixels per side; unchanged coefficient pairs and sizes reuse their image. Model/field/selection changes preserve an open density workspace. Options contains exact shifts and the separate 3Dorbital viewer. Reset restores default fields, model and states and returns to levels. Only raw E/B text, model and selected state IDs are retained, including blank, invalid, comma or inactive-field drafts. Legacy numeric inputs migrate. Plots, raster images and an open density workspace are never restored;reload begins with live levels.

Ideal Coulomb hydrogen, infinite nuclear mass, spin omitted, first-order perturbation within n=2. In basis (2s,2p₀,2p₊,2p₋), the electric perturbation eEz has only off-diagonal s–p₀ entries −3ea₀E; the orbital magnetic perturbation has diagonal entries (0,0,+μBB,−μBB). E in kV/cm is multiplied by10⁵ to get V/m; B is in tesla. Combined shifts are therefore ±3a₀E and ±μBB, not pairwise sums. S±=(2s∓2p₀)/√2 have shifts ±3a₀E. With E=0, the spherical s/p₀ basis is used; degenerate bases are not unique.

Signs of E and B refer to the common z axis. For nonparallel fields this Hamiltonian is insufficient. No fine/hyperfine structure, Lamb shift, electron spin, quadratic Stark/diamagnetic terms, higher-n mixing or field ionization. This ideal first-order manifold is not a precision prediction for real hydrogen at every allowed field, especially close to zero where neglected splittings matter.

The spectrum uses shifts δE from the unperturbed n=2 energy, avoiding cancellation of small shifts against a large binding energy. |ΔE| and λ=hc/|ΔE| are conversions, not proof of an observed transition. E1 is classified using the projected electric-dipole matrix, normalized to9a₀². Strength1 corresponds to a pure s–p pair; a Stark mixture to p± has1/2. S+↔S− and p+↔p− have zero E1 coupling in this truncated model. π denotes z coupling, σ transverse coupling. Strength is not an emission rate; polarization, populations and frequency factors are not included. Zero gap has no finite-frequency photon.

Density uses ψ=C exp(−r/2)[cₛ(2−r)+c₀z+(−c₊+c₋)x/√2−i(c₊+c₋)y/√2], C=1/√(32π), coordinates in a₀. Displayed |ψ|² has units a₀⁻³; the y=0 section is not an integrated probability. The two images share a logarithmic colour scale. Finite image extent and basis functions describe first-order projected states, not exact field-distorted orbitals.

Richard Fitzpatrick: Linear Stark Effect · MIT: Hydrogen Fine Structure