An electron has orbital quantum number l = 2 and spin s = 1/2. What are the possible values of the total angular momentum quantum number j?
Aj = 5/2 only, since j = l + s is the maximum total
Bj = 3/2 and j = 5/2, since j ranges from |l − s| to l + s in integer steps
Cj = 2, 3/2, 1, 1/2 — all half-integer values from 1/2 up to l + s
Dj = l × s = 1, since quantum numbers combine multiplicatively
The total angular momentum quantum number j takes all values from |l − s| to l + s in integer steps. For l = 2, s = 1/2: |l − s| = |2 − 1/2| = 3/2 and l + s = 5/2. The integer-step rule gives j = 3/2 and j = 5/2 — exactly two values. The j = 3/2 level has 2j+1 = 4 magnetic substates and j = 5/2 has 6, totaling 10 states — matching the (2l+1)(2s+1) = 5×2 = 10 states in the uncoupled basis, confirming the two descriptions span the same space.
Question 2 Multiple Choice
When spin-orbit coupling is significant in an atom, why do physicists prefer the coupled basis |j, mⱼ⟩ over the uncoupled basis |mₗ, mₛ⟩?
ABecause mₗ and mₛ are not measurable in principle for any real atomic state
BBecause spin-orbit coupling means L and S are no longer separately conserved — J is the good conserved quantity, making |j, mⱼ⟩ the natural eigenbasis for the Hamiltonian
DBecause the uncoupled basis |mₗ, mₛ⟩ violates the Pauli exclusion principle for electrons
The spin-orbit interaction H_SO ∝ L⃗·S⃗ does not commute with Lz or Sz individually, meaning mₗ and mₛ are no longer good quantum numbers — they are not conserved. However, H_SO does commute with J², Jz, L², and S², so j and mⱼ (along with l and s) are good quantum numbers. When solving the hydrogen fine structure, one works in the coupled basis because those are the actual energy eigenstates. The uncoupled basis is used when spin-orbit coupling is negligible (e.g., weak external magnetic field dominating — the Paschen-Back regime).
Question 3 True / False
For an electron in an s orbital (l = 0), the only possible total angular momentum quantum number is j = 1/2.
TTrue
FFalse
Answer: True
The range of j is from |l − s| to l + s in integer steps. For l = 0, s = 1/2: |0 − 1/2| = 1/2 and 0 + 1/2 = 1/2. The only value is j = 1/2. This means an s-orbital electron has total angular momentum entirely from its spin, with no orbital contribution to j. The two magnetic substates are mⱼ = +1/2 and mⱼ = −1/2.
Question 4 True / False
The coupled basis |j, mⱼ⟩ and the uncoupled basis |mₗ, mₛ⟩ span Hilbert spaces of different dimensions, since combining angular momenta changes the number of accessible quantum states.
TTrue
FFalse
Answer: False
Both bases span exactly the same Hilbert space with the same number of dimensions: (2l+1)(2s+1). The coupled and uncoupled bases are two different ways of spanning the same space — like two different coordinate systems in the same vector space. For l = 1, s = 1/2: the uncoupled basis has (3)(2) = 6 states. The coupled basis has j = 3/2 (4 states) and j = 1/2 (2 states) — again 6 total. The Clebsch-Gordan coefficients are precisely the unitary transformation between these two bases.
Question 5 Short Answer
Why is the total angular momentum J⃗ = L⃗ + S⃗ the conserved quantity in spin-orbit coupling, rather than L⃗ and S⃗ separately? What physically breaks the separate conservation?
Think about your answer, then reveal below.
Model answer: Spin-orbit coupling arises from the interaction between an electron's magnetic moment (from its spin) and the magnetic field it experiences due to its orbital motion around the nucleus. This interaction is proportional to L⃗·S⃗. Because this term mixes L and S, neither L nor S is individually conserved under time evolution — their directions precess around the total J⃗. However, J⃗ = L⃗ + S⃗ commutes with the full Hamiltonian (including H_SO), so J⃗ is conserved. Physically, L and S are coupled by the interaction and exchange angular momentum between them, while J (their sum) is the quantity the system conserves. The magnitudes L² and S² (and hence l and s) are still conserved; only the z-components Lz and Sz are not.
This is a general principle in quantum mechanics: when two systems interact, individually conserved quantities can cease to be conserved, but the total of the coupled system often remains conserved by symmetry. Spin-orbit coupling is the physical mechanism behind atomic fine structure — the splitting of spectral lines into doublets and multiplets. The sodium D-line doublet, for example, arises from the j = 3/2 and j = 1/2 levels of the 3p electron (l = 1), with the energy splitting proportional to the L⃗·S⃗ expectation value in each state.