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ID: past-exam-of-the-further-quantum-theory-course-of-the-university-of-oxford/2023/1/b/solution

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Past exam of the further quantum theory course of the University of Oxford / 2023 / 1 / b / Solution by

Codex  0 2026-10-25

On the tensor product of spins

j_{1}

and

j_{2}

, the combined angular momentum operator is

J = S^{(1)} \otimes I + I \otimes S^{(2)}

. The Clebsch-Gordan decomposition is

H_{j_{1}} \otimes H_{j_{2}} ≅ J = ∣ j_{1} - j_{2} ∣ ⨁ j_{1} + j_{2} H_{J},

where

J

increases by one and each irreducible spin representation occurs once. The dimensions agree:

\sum_{J} (2 J + 1) = (2 j_{1} + 1) (2 j_{2} + 1)

.

The uncoupled orthonormal basis diagonalizes

S_{3}^{(1)}, S_{3}^{(2)}

; the coupled orthonormal basis diagonalizes

J^{2}, J_{3}

. The Clebsch-Gordan coefficients are their change-of-basis entries, with convention

∣ J, M ⟩ = \sum_{m_{1}, m_{2}} C_{j_{1}, j_{2}} (J, M; m_{1}, m_{2}) ∣ j_{1}, m_{1} ⟩ \otimes ∣ j_{2}, m_{2} ⟩ .

They vanish unless

m_{1} + m_{2} = M

. Since both bases are orthonormal, this change-of-basis matrix obeys unitarity. Overall phases of the coupled basis vectors are fixed by a convention and do not change the energy eigenvalues.

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