Questions: Fine Structure and Relativistic Corrections

Question 1

Which two physical effects contribute comparably to the fine structure of hydrogen, lifting the degeneracy between states with the same n but different l?

• A. Zeeman effect and Lamb shift
• B. Relativistic kinetic energy correction and spin-orbit coupling
• C. Nuclear spin coupling and vacuum polarization
• D. Diamagnetic correction and hyperfine interaction

Answer: (B)

Fine structure has two comparably-sized contributions. First, the relativistic kinetic energy correction: expanding the full relativistic kinetic energy to order (v/c)² yields a p⁴ term that depends on both n and l, breaking l-degeneracy. Second, spin-orbit coupling: the electron's magnetic moment interacts with the magnetic field it 'sees' as the proton orbits it in the electron's rest frame; this adds an L⃗·S⃗ term that also depends on l. The Zeeman effect requires an external field; the Lamb shift is a QED correction beyond fine structure; hyperfine structure comes from nuclear spin and is ~1000× smaller still.

Question 2

A student says the fine structure states of hydrogen are labeled by quantum numbers n, l, m_l, and m_s. What is wrong with this description?

• A. Nothing — those are exactly the right quantum numbers for fine structure states
• B. Fine structure only requires n and l; spin plays no role
• C. The correct quantum numbers are n, l, j, and m_j — because spin-orbit coupling makes m_l and m_s individually non-conserved
• D. The correct quantum numbers are n and j only; l is no longer defined once spin-orbit coupling is included

Answer: (C)

Once spin-orbit coupling (L⃗·S⃗) is added to the Hamiltonian, L⃗ and S⃗ precess around their sum J⃗ = L⃗ + S⃗. The operators L_z and S_z no longer commute with the full Hamiltonian, so m_l and m_s are not individually conserved. The good quantum numbers become n (principal), l (orbital magnitude, still conserved), j = l ± ½ (total angular momentum magnitude), and m_j (total z-projection). Spectroscopic notation reflects this: the 2p levels become 2P₃/₂ and 2P₁/₂, labeled by j not by m_l and m_s.

Question 3

Fine structure alone predicts that the 2S₁/₂ and 2P₁/₂ states of hydrogen are degenerate — they have the same energy within the fine structure approximation.

• T. True
• F. False

Answer: True

Fine structure energy corrections depend on n and j (total angular momentum). Both 2S₁/₂ (n=2, l=0, j=½) and 2P₁/₂ (n=2, l=1, j=½) have the same n=2 and j=½, so within Dirac theory (which accounts for relativistic corrections and spin-orbit coupling), they are degenerate. Their actual separation — the Lamb shift of ~1058 MHz — is a quantum electrodynamic effect from vacuum fluctuations and electron self-energy, which lies beyond fine structure. The Lamb shift was a key experimental triumph for QED.

Question 4

Fine structure energy corrections are comparable in magnitude to the gross structure (Bohr) energy level spacings, which is why they are visible in ordinary spectroscopy.

• T. True
• F. False

Answer: False

Fine structure corrections scale as α² × Eₙ, where α ≈ 1/137 is the fine structure constant. Since α² ≈ 5×10⁻⁵, fine structure corrections (~10⁻³ eV) are roughly 10,000 times smaller than the gross structure spacings (~1–10 eV). This is why fine structure is 'fine' — it requires high-resolution spectroscopy to resolve. The sodium D-line doublet, one of the most famous examples, consists of two lines only 0.6 nm apart (589.0 and 589.6 nm); at low resolution they appear as a single yellow line.

Question 5

Why do m_l and m_s cease to be good quantum numbers when spin-orbit coupling is added to the hydrogen Hamiltonian, and what replaces them?

Model answer: Spin-orbit coupling adds a term proportional to L⃗·S⃗ to the Hamiltonian. This operator does not commute with L_z or S_z individually, so those quantities are no longer conserved. Instead, L⃗ and S⃗ precess around the total J⃗ = L⃗ + S⃗, and the conserved quantities become the magnitudes of L⃗, S⃗, and J⃗ (quantum numbers l, s, j) and the z-projection of the total angular momentum m_j.

A quantum number is 'good' — a constant of the motion — if and only if its operator commutes with the Hamiltonian. Writing L⃗·S⃗ = ½(J² − L² − S²) shows it commutes with J², L², and S² but not with L_z or S_z. So after including spin-orbit coupling, the set of commuting observables that diagonalizes the Hamiltonian is {H, L², S², J², J_z}, corresponding to quantum numbers {n, l, s=½, j, m_j}. The individual z-projections m_l and m_s fluctuate because L⃗ and S⃗ are coupled and precess.