An electron in a hydrogen atom has orbital quantum number l = 1 and spin s = 1/2. What are the allowed values of total angular momentum quantum number j?
Aj = 0, 1/2, 1, 3/2, and 2 — all values from 0 to l + s
Bj = 1/2 and j = 3/2 — the values from |l − s| to l + s in integer steps
Cj = 3/2 only — angular momenta always add to their maximum
Dj = 1/2, 1, and 3/2 — all half-integer and integer values between |l − s| and l + s
The rule is |j₁ − j₂| ≤ j ≤ j₁ + j₂ in integer steps. With l = 1 and s = 1/2: |1 − 1/2| = 1/2 and 1 + 1/2 = 3/2. The allowed values are j = 1/2 and j = 3/2 — exactly two values, separated by 1. The total number of states is (2×1/2+1) + (2×3/2+1) = 2 + 4 = 6, which equals (2l+1)(2s+1) = 3 × 2 = 6. The intermediate values (j = 1, j = 0) are not allowed here because they don't fit the integer-step rule starting from |l − s| = 1/2.
Question 2 Multiple Choice
For two spin-1/2 particles, the uncoupled state |↑↓⟩ (particle 1 up, particle 2 down) is not an eigenstate of J². A measurement of J² on this state would:
AAlways give j = 1, since the spins are anti-aligned and the triplet includes an m = 0 state
BAlways give j = 0, since the spins cancel
CGive j = 1 or j = 0 with certain probabilities, because |↑↓⟩ is a superposition of triplet and singlet states
DBe undefined, because |↑↓⟩ is not a valid quantum state for coupled angular momenta
|↑↓⟩ can be written as a superposition of the triplet m=0 state (|↑↓⟩ + |↓↑⟩)/√2 and the singlet state (|↑↓⟩ − |↓↑⟩)/√2: specifically, |↑↓⟩ = [(triplet m=0) + (singlet)]/√2. Measuring J² collapses this superposition, yielding j = 1 with probability 1/2 and j = 0 with probability 1/2. This illustrates why the uncoupled basis is inconvenient when J² matters — neither |↑↓⟩ nor |↓↑⟩ is a total-angular-momentum eigenstate.
Question 3 True / False
When two angular momenta j₁ and j₂ are coupled, the total number of states in the coupled basis equals (2j₁+1)(2j₂+1), the same dimension as the uncoupled basis.
TTrue
FFalse
Answer: True
True. The coupled and uncoupled bases are two different orthonormal bases for the same Hilbert space — they span the same space. The dimension must be preserved. You can verify: summing (2j+1) over j from |j₁−j₂| to j₁+j₂ in integer steps always gives (2j₁+1)(2j₂+1). For the two spin-1/2 case: (2×1+1) + (2×0+1) = 3 + 1 = 4 = 2×2. The Clebsch-Gordan transformation is unitary precisely because it maps between two orthonormal bases of the same space.
Question 4 True / False
In the coupled basis |j, m⟩, the total z-projection m is no longer a good quantum number — it becomes indefinite because coupling mixes states with different m₁ and m₂ values.
TTrue
FFalse
Answer: False
False. The total z-projection m = m₁ + m₂ remains a good quantum number in both bases. In the uncoupled basis, m₁ and m₂ are individually definite, so their sum is definite. In the coupled basis, m is still definite as the eigenvalue of J_z = J₁z + J₂z. What becomes indefinite in the uncoupled basis is J² (the total magnitude). What the coupling accomplishes is making J² definite while keeping m definite — switching from (j₁, m₁, j₂, m₂) as quantum numbers to (j₁, j₂, j, m).
Question 5 Short Answer
Why is the coupled basis |j, m⟩ more physically useful than the uncoupled basis |j₁m₁⟩|j₂m₂⟩ for a hydrogen atom electron subject to spin-orbit coupling?
Think about your answer, then reveal below.
Model answer: Spin-orbit coupling adds a term to the Hamiltonian proportional to L·S, which equals (J² − L² − S²)/2. This operator is diagonal in the coupled basis |j, m⟩ — states of definite j are energy eigenstates of the spin-orbit perturbation — but mixes states in the uncoupled basis. The energy shift depends on j: the j = l+1/2 and j = l−1/2 levels are split by an amount proportional to the spin-orbit coupling constant. Calculating this splitting requires using the coupled basis where J² is a good quantum number. Spectroscopic selection rules (which transitions are allowed) are also written in terms of j rather than l and s separately.
More generally, any perturbation that depends on total angular momentum (spin-orbit coupling, hyperfine interaction, response to external fields in certain regimes) is most naturally analyzed in the coupled basis. The uncoupled basis is simpler to construct, but the coupled basis reveals the physical symmetry of the problem.