A electron has orbital angular momentum and spin

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 dd electron has orbital angular momentum j1=l=2j_1=l=2 and spin j2=s=12j_2=s=\tfrac12. Use the uncoupled basis ∣ml⟩∣ms⟩|m_l\rangle|m_s\rangle (ml=2,1,0,−1,−2m_l=2,1,0,-1,-2; ms=12,−12m_s=\tfrac12,-\tfrac12) 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=ml+msM=m_l+m_s over all ten product states, peel off multiplets from the top, and derive the decomposition l⊗s=⨁JJl\otimes s=\bigoplus_J J. Check the dimension count. (b) Starting from the unique top state ∣J=52,M=52⟩=∣ml=2⟩∣ms=12⟩|J=\tfrac52,M=\tfrac52\rangle=|m_l=2\rangle|m_s=\tfrac12\rangle, lower once to obtain ∣52,32⟩|\tfrac52,\tfrac32\rangle as a combination of the two M=32M=\tfrac32 product states. Then use orthogonality, with the Condon-Shortley sign convention that the coefficient of ∣ml=2⟩∣ms=−12⟩|m_l=2\rangle|m_s=-\tfrac12\rangle is positive, to build ∣J=32,M=32⟩|J=\tfrac32,M=\tfrac32\rangle.

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