Engineering & Science
Linear & Quadratic Stark Effect Atomic Electric-Field Splitting Formula
Calculate Johannes Stark’s n = 2 hydrogen first-order linear electric-field energy splitting delta_E = 3 * e * a_0 * E (meV) and ground-state polarizability second-order quadratic Stark shift.
Why degenerate excited Hydrogen n = 2 splits linearly in an electric field while n = 1 shifts quadratically!
Most non-degenerate atomic ground states possess symmetric parity and zero permanent electric dipole moment, so an external electric field E (MV/m = 10^6 V/m) must first polarize the electron cloud (p_ind = alpha_p * E) before lowering its energy quadratically via the Second-Order Stark Shift: delta_E_1s = -(9 / 4) * (4 * pi * eps_0 * a_0^3) * E^2!
Johannes Stark (1913, Nobel Prize 1919) discovered a dramatic exception in excited Hydrogen n = 2: because the spherical 2s and dumbbell 2p_0 orbitals are degenerate in energy with opposite parity, an external electric field mixes them into permanent-dipole parabolic eigenstates (|2s> +/- |2p_0>)/SQRT(2) whose outer Stark wings split LINEARLY with electric field: delta_E_wing = +/- 3 * e * a_0 * E (total outer span = 6 * e * a_0 * E)! Since Bohr radius a_0 = 0.0529177 nm, at E = 10 MV/m (100 kV/cm), each outer n = 2 Stark wing shifts by 3 * 0.0529177 * 10 = 1.5875 meV (full doublet span = 3.1751 meV)—scaling linearly to 3.9688 meV at E = 25 MV/m!
| Electric Field Strength E (MV/m) | Hydrogen n = 2 Single-Wing Shift 3*e*a0*E (meV) | Total Outer Span 6*e*a0*E (meV) | Quantum Parity Regime |
|---|---|---|---|
| 10 MV/m (100 kV/cm Stark Tube) | 1.5875 meV | 3.1751 meV | Degenerate 2s-2p Linear Mixing |
| 25 MV/m (250 kV/cm High-Field) | 3.9688 meV (2.5x Linear) | 7.9377 meV | Below n = 2 Tunneling Ionization |
| Negative Electric Field Magnitude < 0 | Guarded Input Error | "Invalid Electric Field" | Requires E >= 0 |
Hydrogen n = 2 First-Order Linear Stark Wing Shift delta_E (meV) = 3 * 0.0529177 * E_MVm
Since 1 nm * 1 MV/m = 10^-9 m * 10^6 V/m = 10^-3 V = 1 mV, multiplying 3 * a_0_nm (0.0529177) by E (MV/m, B2) divided by nuclear charge Z (B3 = 1) gives exact meV!
=IF(OR(B2 < 0, B3 < 1), "Invalid Electric Field", ROUND((3 * 0.0529177 * B2) / B3, 4))Full Outer Stark Doublet Span 2 * delta_E = 6 * e * a_0 * E / Z (meV)
Computes the total energy separation between the upper (+3ea0E) and lower (-3ea0E) parabolic n = 2 manifolds.
=IF(OR(B2 < 0, B3 < 1), "Invalid Electric Field", ROUND((6 * 0.0529177 * B2) / B3, 4))Try the example
Use the original values or change them to test your understanding.