The ground state of the nucleus has spin

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Group theory and symmetry · Group theory and symmetry

Invariance, Ch Representations of Lie algebras: spin, products and SU(3)

Problem

The ground state of the 7Li^7\mathrm{Li} nucleus has spin j=32j=\tfrac32, so it is described by the four-dimensional highest-weight representation of su(2)\mathfrak{su}(2) with basis states ∣j,m⟩|j,m\rangle, m=32,12,−12,−32m=\tfrac32,\tfrac12,-\tfrac12,-\tfrac32, satisfying J+∣j,j⟩=0J_+|j,j\rangle=0. ((a)) Using J2=J−J++Jz2+JzJ^2=J_-J_++J_z^2+J_z acting on the top state ∣32,32⟩|\tfrac32,\tfrac32\rangle, evaluate the Casimir eigenvalue. ((b)) State the value of j(j+1)j(j+1) and confirm it agrees with your result.

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