Questions: Selection Rules for Atomic Transitions

Question 1

A hydrogen atom is in the 2s excited state (n=2, ℓ=0). Why doesn't it rapidly decay to the 1s ground state (n=1, ℓ=0) via electric dipole radiation?

• A. The energy difference between 2s and 1s is too small to produce a detectable photon
• B. The 2s → 1s transition requires Δℓ = 0, which violates the electric dipole selection rule Δℓ = ±1 — the emitted photon must carry 1ℏ of angular momentum, which the atom cannot supply with this transition
• C. The 2s state has no orbital angular momentum, so it cannot couple to the electromagnetic field at all
• D. The 2s → 1s transition is forbidden because the principal quantum number change Δn = 1 is too small

Answer: (B)

Angular momentum must be conserved. A photon carries 1ℏ of angular momentum, so the atom must change its orbital angular momentum by exactly 1ℏ — requiring Δℓ = ±1. The 2s state has ℓ = 0 and the 1s state has ℓ = 0, so Δℓ = 0: the atom cannot supply the angular momentum the photon needs. The transition is electric dipole forbidden. The 2s state is therefore metastable, with a much longer lifetime than states that can decay via allowed transitions.

Question 2

Why do forbidden spectral lines appear prominently in emission nebulae but are essentially unobservable in laboratory plasmas at similar temperatures?

• A. Nebulae contain different elements than laboratory plasmas, so the forbidden lines come from exotic atoms not present in the lab
• B. In nebulae, particle densities are so low that atoms in metastable states decay via the slow forbidden transitions before collisions can de-excite them; in lab plasmas, collisions occur far faster than the radiative decay
• C. The high magnetic fields in nebulae relax the selection rules, allowing normally forbidden transitions to proceed rapidly
• D. Forbidden lines are at infrared wavelengths that ground-based telescopes detect but lab spectrometers cannot

Answer: (B)

Forbidden transitions have radiative lifetimes roughly 10⁵–10⁸ times longer than allowed transitions — milliseconds to hours instead of nanoseconds. In a lab plasma, collisions happen on timescales of microseconds or less, so an atom in a metastable state is almost always collisionally de-excited before it can radiate. In nebulae, number densities may be 10⁴ particles/cm³ or less (vs. ~10¹⁹/cm³ in lab air), and the mean time between collisions is long enough that the slow radiative decay wins. The green lines of [O III] in planetary nebulae are a famous example.

Question 3

A transition labeled 'electric dipole forbidden' cannot occur under any circumstances.

• T. True
• F. False

Answer: False

Electric dipole forbidden means the dominant electric dipole mechanism is unavailable — not that the transition is impossible. Weaker mechanisms (magnetic dipole, electric quadrupole, and higher multipoles) can still drive the transition, though they are roughly 10⁵–10⁸ times slower. 'Forbidden' in spectroscopy is a relative term: it names the suppression of one specific mechanism, not an absolute prohibition. Metastable states in nebulae exploit exactly this: the weak forbidden mechanism wins when collisions are rare enough.

Question 4

The selection rule Δℓ = ±1 for electric dipole transitions arises from conservation of angular momentum: an emitted photon carries angular momentum of exactly 1ℏ, which must be supplied by a change in the atom's orbital angular momentum.

• T. True
• F. False

Answer: True

This is the physical origin of the rule. Photons are spin-1 particles carrying angular momentum of 1ℏ. Total angular momentum (atom + photon) must be conserved during emission, so the atom's angular momentum must change by ±1ℏ. Since ℓ characterizes orbital angular momentum in units of ℏ, this requires Δℓ = ±1. The mathematical formalism — that the transition matrix element ⟨f|r|i⟩ vanishes unless initial and final states have opposite parity, requiring odd Δℓ — implements the same physical constraint.

Question 5

Explain the physical origin of the Δℓ = ±1 selection rule. Why does the 2s → 1s transition in hydrogen violate it, and what is the physical fate of hydrogen atoms trapped in the 2s metastable state in a very low-density environment?

Model answer: The selection rule arises from angular momentum conservation: photons carry 1ℏ of angular momentum, so the atom must change its orbital angular momentum by 1ℏ, requiring Δℓ = ±1. The 2s → 1s transition has Δℓ = 0 (both states have ℓ = 0), so it cannot proceed via electric dipole radiation — the atom cannot supply the angular momentum the photon requires. In a very low-density environment, a hydrogen atom in the 2s state can eventually decay by simultaneously emitting two photons (two-photon decay), which share the energy and angular momentum between them, allowing Δℓ = 0 with no single-photon angular momentum constraint. The 2s state has a lifetime of about 0.12 seconds via this mechanism — enormously long compared to typical allowed transitions (~nanoseconds).

The two-photon decay of the 2s state is astrophysically important: it contributes to the diffuse emission of hydrogen nebulae and was historically significant for understanding atomic physics. The key conceptual point is that 'forbidden' means 'forbidden via the dominant mechanism' — exotic alternative decay channels always exist but are much slower.