A composite system has two subsystems with angular momentum quantum numbers j₁ = 3/2 and j₂ = 1. What are the possible values of the total angular momentum quantum number j?
Aj = 1/2, 3/2, 5/2
Bj = 0, 1/2, 1, 3/2, 2, 5/2
Cj = 5/2 only — the maximum total angular momentum
Dj = 1/2, 1, 3/2, 2, 5/2 — all values between the minimum and maximum
The triangular rule gives j from |j₁ − j₂| to j₁ + j₂ in integer steps: |3/2 − 1| = 1/2 and 3/2 + 1 = 5/2, so j = 1/2, 3/2, 5/2. Option B is the common error — assuming all values from 0 up to j₁ + j₂ are allowed. The lower bound is |j₁ − j₂|, not 0. Dimension check confirms: (2·½+1) + (2·3/2+1) + (2·5/2+1) = 2 + 4 + 6 = 12 = (2·3/2+1)(2·1+1) = 4·3 ✓.
Question 2 Multiple Choice
An atom has spin-orbit coupling described by a term proportional to L·S in the Hamiltonian. Why is the coupled basis {|j, M⟩} preferred over the uncoupled basis {|mₗ, mₛ⟩} for computing energy levels?
AThe coupled basis is mathematically simpler for all Hamiltonians, regardless of the physical interaction
BL·S commutes with J² and Jz but not with Lz or Sz individually, so j and M are good quantum numbers while mₗ and mₛ are not
CThe uncoupled basis fails to span the full state space when spin-orbit coupling is present
DThe coupled basis eliminates the need for Clebsch-Gordan coefficients once the basis change is made
L·S = (J² − L² − S²)/2, which commutes with J², Jz, L², and S² but not with Lz or Sz. This means j and M are conserved quantities (good quantum numbers) under spin-orbit coupling, while mₗ and mₛ are not. In the coupled basis, L·S is diagonal, yielding energy eigenvalues directly. In the uncoupled basis, spin-orbit coupling mixes states, requiring diagonalization. The physical energy splitting (fine structure) is indexed by j, not by mₗ and mₛ separately.
Question 3 True / False
When combining two spin-1/2 particles, the possible values of the total spin quantum number include j = 0, 1/2, and 1.
TTrue
FFalse
Answer: False
Applying the triangular rule: |1/2 − 1/2| = 0 and 1/2 + 1/2 = 1, with integer steps, gives j = 0 and j = 1 only. The value j = 1/2 is not allowed. This is a common error — students assume all values between 0 and the maximum are accessible, but the rule requires integer steps from |j₁ − j₂|, not all fractions. The result is a triplet (j = 1, three states M = −1, 0, +1) and a singlet (j = 0, one state M = 0), totaling 4 = 2×2 states.
Question 4 True / False
The coupled basis and uncoupled basis for a composite angular momentum system span the same Hilbert space and have the same total number of states.
TTrue
FFalse
Answer: True
Both are complete orthonormal bases for the same (2j₁+1)(2j₂+1)-dimensional tensor product space. The Clebsch-Gordan coefficients are the entries of the unitary transformation connecting them — no states are created or lost by changing basis. This is why the dimension count always works out: the sum Σⱼ(2j+1) over all allowed j values must equal (2j₁+1)(2j₂+1).
Question 5 Short Answer
State the triangular rule for combining angular momenta j₁ and j₂, and explain why the minimum total angular momentum is |j₁ − j₂| rather than 0.
Think about your answer, then reveal below.
Model answer: The total angular momentum j ranges from |j₁ − j₂| to j₁ + j₂ in integer steps. The minimum is |j₁ − j₂|, not 0, because angular momenta add vectorially and must obey the triangle inequality. When j₁ ≠ j₂, the larger angular momentum always has a component that the smaller cannot cancel, so j = 0 is impossible. Only when j₁ = j₂ can the two exactly oppose to give j = 0.
The classical analogy is clear: two vectors of lengths 3 and 4 combine to give a resultant between 1 and 7, never 0, because the longer vector has more magnitude. The quantum triangular rule is the analog of the classical triangle inequality, with the additional quantization constraint that j must be integer or half-integer. The dimension check — verifying that state counts are preserved — is a reliable way to confirm the allowed j values.