Questions: Clebsch-Gordan Coefficients

Question 1

An orbital electron has orbital angular momentum j₁ = 1 and spin j₂ = 1/2. What are the possible values of total angular momentum J?

• A. J = 1 only, because spin-½ is a small perturbative correction
• B. J = 3/2 and J = 1/2
• C. J = 2 and J = 1
• D. J = 1/2 only, because spin dominates the coupling

Answer: (B)

When combining angular momenta j₁ and j₂, total J ranges in integer steps from |j₁ − j₂| to j₁ + j₂. Here: |1 − 1/2| = 1/2 and 1 + 1/2 = 3/2, giving J = 3/2 (4 states) and J = 1/2 (2 states). The total count is 4 + 2 = 6 = (2×1+1)(2×1/2+1) = 3 × 2, as required. Options A and D incorrectly treat one angular momentum as dominant; they always combine vectorially, with all values between the min and max allowed.

Question 2

You want to expand |j₁=1, m₁=1; j₂=1/2, m₂=−1/2⟩ in the coupled basis. Which coupled states |J, M⟩ can appear in this expansion?

• A. Any state with J between 1/2 and 3/2
• B. Only states with M = m₁ + m₂ = +1/2, specifically |J=3/2, M=1/2⟩ and |J=1/2, M=1/2⟩
• C. Only |J=3/2, M=1/2⟩ because the higher-J state is dominant
• D. States with any M value, since the full expansion sums over all coupled states

Answer: (B)

The most important CG selection rule is M = m₁ + m₂. The z-components of angular momentum add, so an uncoupled state |m₁, m₂⟩ can only contribute to coupled states with M = m₁ + m₂ = 1 + (−1/2) = +1/2. Both J = 3/2 and J = 1/2 have M = +1/2 states, so both appear with nonzero coefficients. The actual coefficients require looking up CG tables, but the selection rule immediately eliminates all M ≠ +1/2 states — most of the table.

Question 3

The selection rule M = m₁ + m₂ in Clebsch-Gordan decomposition reflects the fact that the z-component of total angular momentum equals the sum of the individual z-components.

• T. True
• F. False

Answer: True

The z-component operator is Jz = J₁z + J₂z. The uncoupled state |j₁, m₁; j₂, m₂⟩ is an eigenstate of J₁z with eigenvalue ℏm₁ and of J₂z with eigenvalue ℏm₂, so it is an eigenstate of Jz with eigenvalue ℏ(m₁ + m₂). Since coupled states |J, M⟩ are eigenstates of Jz with eigenvalue ℏM, only coupled states with M = m₁ + m₂ are non-orthogonal to the uncoupled state. This is an exact selection rule — not an approximation — and it eliminates most CG coefficients from the outset.

Question 4

For two spin-½ particles, the Clebsch-Gordan decomposition yields four total states, most of which are symmetric under exchange of particle labels.

• T. True
• F. False

Answer: False

The decomposition yields a triplet (J=1) and a singlet (J=0). The three triplet states |1,1⟩, |1,0⟩, |1,−1⟩ are symmetric under particle exchange, but the singlet |0,0⟩ = (1/√2)(|↑↓⟩ − |↓↑⟩) is antisymmetric — it reverses sign when particle labels are swapped. The CG coefficient structure encodes this symmetry difference: a + sign in the triplet M=0 state, a − sign in the singlet. This matters because identical fermions require antisymmetric total wavefunctions, constraining which spin states are available for a given spatial state.

Question 5

What does the Clebsch-Gordan coefficient ⟨j₁, m₁; j₂, m₂ | J, M⟩ physically represent, and why does the selection rule M = m₁ + m₂ guarantee that most of these coefficients are exactly zero?

Model answer: The coefficient ⟨j₁, m₁; j₂, m₂ | J, M⟩ is the inner product between the uncoupled state |j₁, m₁; j₂, m₂⟩ and the coupled state |J, M⟩. It gives the probability amplitude for finding the system in the coupled state |J, M⟩ when the particles are individually in states |m₁⟩ and |m₂⟩. The selection rule M = m₁ + m₂ follows from the additivity of Jz: any coupled state with M ≠ m₁ + m₂ is a Jz eigenstate with a different eigenvalue and is therefore orthogonal to the uncoupled state, making the inner product exactly zero.

The interpretation as probability amplitudes makes CG coefficients central to any quantum measurement involving composite angular momentum. If you prepare two particles in |m₁, m₂⟩ and measure total J, the probability of obtaining J = 3/2 is |⟨j₁, m₁; j₂, m₂ | 3/2, m₁+m₂⟩|². The Wigner-Eckart theorem extends this further: matrix elements of tensor operators between angular momentum states factor into a CG coefficient times a reduced matrix element, which is why CG tables appear in nearly every spectroscopy and atomic physics calculation.