Quantum Mechanics (1) · Problem 16

Identical Particles

Physics — Quantum Mechanics · clebsch-gordan-coefficients

Problem

It is possible to arrive at the Clebsch-Gordan coefficients for a given situation by ``elementary'' means, i.e., by considering the action of the raising and lowering operators and demanding orthonormality. Hence, construct a table of Clebsch-Gordan coefficients, using this approach, for a system combining j1=2j_1 = 2 and j2=1j_2 = 1 angular momenta. [I find it convenient to use the simple notation ∣jm⟩|jm\rangle for total quantum numbers and ∣j1m1⟩∣j2m2⟩|j_1 m_1\rangle|j_2 m_2\rangle for the individual angular momentum states being added, but you may use whatever notation you find convenient.] You will find (I hope) that you have the freedom to pick certain signs. You are asked to be consistent with the usual conventions where ⟨33∣(∣22⟩∣11⟩)≥0⟨22∣(∣22⟩∣10⟩)≥0⟨11∣(∣22⟩∣1−1⟩)≥0(5.528)\begin{aligned} \langle 33|(|22\rangle|11\rangle) &\geq 0 \tag{5.528}\\ \langle 22|(|22\rangle|10\rangle) &\geq 0 \\ \langle 11|(|22\rangle|1{-}1\rangle) &\geq 0 \end{aligned} (in notation ⟨jm∣(∣j1m1⟩∣j2m2⟩)\langle jm|(|j_1 m_1\rangle|j_2 m_2\rangle)).

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