Questions: Fine Structure: Spin-Orbit Coupling and Doublet Splitting

Question 1

If the Thomas precession correction (factor of 1/2) were neglected in deriving the spin-orbit Hamiltonian, what would the predicted fine-structure splitting in hydrogen be?

• A. Half the observed splitting — the Thomas factor doubles the interaction strength
• B. Twice the observed splitting — without the factor of 1/2, the predicted splitting would be twice what is measured
• C. The same — Thomas precession is a minor correction that does not affect the energy levels
• D. Zero — Thomas precession is what generates the spin-orbit interaction in the first place

Answer: (B)

A naive derivation of spin-orbit coupling (working purely in the electron's rest frame without the relativistic Thomas correction) yields H_SO = (e²/2m²c²r³) S·L, which is twice the correct value. Thomas precession — a purely relativistic kinematic effect arising from the electron's non-inertial (accelerating) rest frame — introduces a compensating factor of 1/2. With the Thomas factor, the Hamiltonian becomes H_SO = (1/2)(e²/2m²c²r³) S·L, matching the observed splitting. The factor of 2 is not a minor correction — omitting it gives predictions that are quantitatively wrong.

Question 2

After including spin-orbit coupling as a perturbation to the hydrogen Hamiltonian, which set of quantum numbers provides valid energy eigenstates for the 2p level?

• A. n, ℓ, mℓ, ms — all four quantum numbers from the unperturbed hydrogen atom remain good
• B. n, ℓ, j, mⱼ — total angular momentum j replaces the separate mℓ and ms as good quantum numbers
• C. n, j, mⱼ only — ℓ is no longer a good quantum number under spin-orbit coupling
• D. n, s, ms — only spin quantum numbers survive as good quantum numbers

Answer: (B)

Spin-orbit coupling mixes states with different mℓ and ms values (S·L does not commute with Lz or Sz individually), so mℓ and ms are no longer good quantum numbers. However, the total angular momentum J = L + S does commute with H_SO, so j and mⱼ are good quantum numbers. The Hamiltonian is diagonalized in the |n, ℓ, j, mⱼ⟩ basis. For the 2p level (ℓ=1, s=1/2), j can be 3/2 or 1/2, giving the 2P₃/₂ and 2P₁/₂ states.

Question 3

The spin-orbit interaction arises because, in the electron's rest frame, the moving nucleus creates a magnetic field that interacts with the electron's spin magnetic moment.

• T. True
• F. False

Answer: True

This is the correct physical mechanism. In the lab frame, the nucleus is stationary and creates only an electric field E. Transforming to the electron's rest frame via special relativity, the moving nucleus produces a magnetic field B ~ v × E/c². This magnetic field interacts with the electron's spin magnetic moment μ_s, with energy −μ_s·B. Expanding this in terms of orbital angular momentum L (since v × E ∝ L for a central potential) gives the S·L coupling term.

Question 4

Spin-orbit coupling is a purely quantum-mechanical effect that has no connection to special relativity — it arises mostly from the intrinsic quantum property of electron spin.

• T. True
• F. False

Answer: False

Spin-orbit coupling is explicitly a relativistic effect. The magnetic field that the electron experiences is a relativistic transformation of the nuclear electric field (v × E/c²); this is zero in the non-relativistic limit. The Thomas precession correction is also purely relativistic, arising from the non-inertial character of the electron's rest frame under Lorentz boosts. The fine structure, of which spin-orbit coupling is a part, emerges from the first-order relativistic corrections to the Schrödinger equation — formalized in the Dirac equation.

Question 5

Why does spin-orbit coupling cause the hydrogen 2p level to split into two distinct energy levels, and what quantum numbers label these two levels?

Model answer: The spin-orbit Hamiltonian is proportional to S·L = (J² − L² − S²)/2. For the 2p level (ℓ=1, s=1/2), the total angular momentum quantum number j can take values j = ℓ + s = 3/2 or j = ℓ − s = 1/2. The expectation value of S·L differs for these two j values — [j(j+1) − ℓ(ℓ+1) − s(s+1)]ℏ²/2 gives different numbers for j = 3/2 and j = 1/2 — so they have different energies. The two levels are labeled 2P₃/₂ and 2P₁/₂.

The splitting is observable in hydrogen's spectrum as a closely spaced doublet. The 2P₃/₂ level (j=3/2) lies higher in energy than 2P₁/₂ (j=1/2) for hydrogen (consistent with the positive spin-orbit coupling constant for ℓ > 0). The magnitude of the splitting matches the relativistic spin-orbit prediction — one of the early experimental confirmations that quantum mechanics and special relativity must be unified, eventually achieved by Dirac's relativistic quantum equation.