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Quantum Mechanics

Stark Effect

The Stark effect is the splitting or shifting of quantum energy levels in an applied electric field. The basic perturbation is electric-dipole coupling, but the observed response depends on parity, degeneracy, polarizability, and whether the field is static or oscillatory.

Formula Hook

Section titled “Formula Hook”

For a static electric field,

VE=−d⋅E.V_E = - \mathbf d\cdot\mathbf E.

For a nondegenerate stationary state ∣n⟩\lvert n\rangle, first-order perturbation theory gives

ΔEn(1)=−⟨n∣d⋅E∣n⟩.\Delta E_n^{(1)} = - \langle n\rvert \mathbf d\cdot\mathbf E \lvert n\rangle.

If parity makes the first-order shift vanish, the leading static shift is often written

ΔEn≈−12αnE2,\Delta E_n \approx - \frac{1}{2}\alpha_n \mathcal E^2,

where αn\alpha_n is the polarizability in the chosen convention.

Canonical Home

Section titled “Canonical Home”

This glossary entry is the named-effect home. Stark Effect as a Perturbation Example owns the static method choice, polarizability check, and ideal-hydrogen calculation. Stark Effect in Atoms owns scalar, vector, and tensor polarizabilities, AC shifts, traps, and spectroscopy. For adjacent examples and concepts, see Rotor in External Fields: First Encounter, Degeneracy Lifting, Nondegenerate Perturbation Theory, and Degenerate Perturbation Theory.

Common Confusions

Section titled “Common Confusions”
  • A first-order Stark shift vanishes for many nondegenerate parity eigenstates, but not for every system.
  • Degenerate hydrogen has a linear Stark effect because the perturbation mixes degenerate opposite-parity states.
  • The AC Stark effect is a driven, frequency-dependent light-shift problem, not just the DC formula with a time-varying field inserted.
  • Polarizability signs and factors of 1/21/2 depend on the energy convention being used.
Section titled “Related Entries”

References

Section titled “References”
  • C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, Wiley, 1977.
  • C. J. Foot, Atomic Physics, Oxford University Press, 2005.
  • D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.