A particle of spin occupies a -wave orbital state with orbital angular momentum

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Symmetry, angular momentum and spin

The Structure of Quantum Mechanics, Ch Adding angular momenta, tensor operators and Wigner-Eckart

Problem

A particle of spin j1=32j_1=\tfrac32 occupies a pp-wave orbital state with orbital angular momentum j2=1j_2=1. Use the uncoupled basis ∣m1⟩∣m2⟩|m_1\rangle|m_2\rangle (m1=32,12,−12,−32m_1=\tfrac32,\tfrac12,-\tfrac12,-\tfrac32; m2=1,0,−1m_2=1,0,-1) and the coupled basis ∣J,M⟩|J,M\rangle, together with J^−∣j,m⟩=ℏj(j+1)−m(m−1) ∣j,m−1⟩\hat J_-|j,m\rangle=\hbar\sqrt{j(j+1)-m(m-1)}\,|j,m-1\rangle and J^−=J^1−+J^2−\hat J_-=\hat J_{1-}+\hat J_{2-}. (a) Tabulate M=m1+m2M=m_1+m_2 over all twelve product states, peel off multiplets from the top, and derive j1⊗j2=⨁JJj_1\otimes j_2=\bigoplus_J J. Check the dimension count. (b) Lower the unique top state ∣J=52,M=52⟩=∣m1=32⟩∣m2=1⟩|J=\tfrac52,M=\tfrac52\rangle=|m_1=\tfrac32\rangle|m_2=1\rangle once to obtain ∣52,32⟩|\tfrac52,\tfrac32\rangle. Then use orthogonality inside the two-dimensional M=32M=\tfrac32 subspace, with the Condon-Shortley sign fixed by requiring the coefficient of ∣m1=32⟩∣m2=0⟩|m_1=\tfrac32\rangle|m_2=0\rangle to be positive, to construct the top state ∣J=32,M=32⟩|J=\tfrac32,M=\tfrac32\rangle of the middle multiplet.

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