Questions: Fine Structure and Relativistic Corrections

Question 1

Fine structure of hydrogen arises from which two relativistic corrections to the non-relativistic Schrödinger equation?

• A. Spin-orbit coupling and the Darwin term (contact interaction)
• B. Relativistic kinetic energy correction and spin-orbit coupling
• C. The Lamb shift and spin-orbit coupling
• D. Hyperfine splitting and the relativistic kinetic energy correction

Answer: (B)

Fine structure comes from two effects: (1) the relativistic correction to kinetic energy — the −p⁴/8m³c² term from expanding relativistic kinetic energy — which preferentially lowers states where the electron has high momentum (small ℓ, close to the nucleus); and (2) spin-orbit coupling, the interaction between the electron's spin magnetic moment and the magnetic field it sees in its rest frame. Together these break the ℓ-degeneracy of the Schrödinger hydrogen atom. The Lamb shift is a QED effect, distinct from fine structure. Hyperfine structure involves nuclear spin and is a separate, much smaller effect.

Question 2

After applying fine-structure corrections to hydrogen, which quantum number correctly distinguishes energy levels within a given principal quantum number n?

• A. The orbital quantum number ℓ alone, since it determines the orbital shape
• B. The total angular momentum quantum number j = ℓ + s, since neither ℓ nor s is individually conserved
• C. The magnetic quantum number mⱼ, since the external field splits levels
• D. Both ℓ and s independently, as separate conserved quantities

Answer: (B)

Spin-orbit coupling is proportional to L·S, which mixes orbital and spin degrees of freedom. Once this term is present, neither L nor S is conserved — only J = L + S is. Therefore j is the good quantum number for fine-structure states, not ℓ or s separately. This is why levels are labeled 2p₁/₂ and 2p₃/₂ — both have ℓ = 1 but j = 1/2 and j = 3/2 respectively. The mⱼ degeneracy is only broken by an external magnetic field (Zeeman effect), which is separate from fine structure.

Question 3

After fine-structure corrections, states with the same n and j have the same energy regardless of ℓ — so 2s₁/₂ and 2p₁/₂ are degenerate at this level of approximation.

• T. True
• F. False

Answer: True

This is the j-degeneracy: the fine-structure energy depends on n and j but not on ℓ separately. The 2s₁/₂ (ℓ=0, j=1/2) and 2p₁/₂ (ℓ=1, j=1/2) states are predicted to be exactly degenerate by fine structure alone. This degeneracy is only lifted by the Lamb shift — a QED effect — which was one of the first great experimental confirmations of quantum electrodynamics. The 2p₃/₂ (j=3/2) level sits higher than both.

Question 4

Hyperfine structure arises from the same physical mechanism as fine structure — both originate in relativistic corrections to the electron's motion.

• T. True
• F. False

Answer: False

Hyperfine structure has a completely different origin: the interaction between the magnetic moment of the electron and the magnetic moment of the nucleus (e.g., the proton's nuclear spin in hydrogen). This is why hyperfine splittings are roughly 1000× smaller than fine-structure splittings — the nuclear magnetic moment is about 1836 times smaller than the electron's due to the proton's much larger mass. Fine structure corrects the electron's own kinetic and magnetic properties; hyperfine structure introduces the nucleus as an active magnetic participant.

Question 5

Why does the fine-structure energy depend on j but not on ℓ and s separately, even though both the relativistic kinetic correction and the spin-orbit term individually depend on ℓ?

Model answer: Both corrections do depend on ℓ individually, but when their contributions are summed, the ℓ-dependence cancels and the combined fine-structure energy depends only on n and j. Using L·S = (J² − L² − S²)/2, the spin-orbit term can be rewritten in terms of j, ℓ, and s. After combining with the kinetic correction (which also depends on ℓ via ⟨p⁴⟩), the total expression simplifies to depend only on n and j — a result that ultimately reflects the structure of the Dirac equation for hydrogen, which produces exact energy levels depending only on n and j. The Schrödinger perturbation calculation reproduces this as a non-trivial cancellation.

This is a hint that the 'right' framework is Dirac's relativistic quantum mechanics, where j is fundamental from the start. The remarkable cancellation between the two fine-structure corrections — leaving j-only dependence — is not a coincidence but reflects the deeper symmetry of the relativistic hydrogen problem. It makes j the natural quantum number and sets the stage for the Lamb shift (a QED correction) as the next important effect.