A toy contact interaction between an electron (spin ) and a positron (spin )
PHYSICS · Multi-part · 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 toy contact interaction between an electron (spin 21) and a positron (spin 21), modelled on the positronium hyperfine interaction, is H^′=ℏ2AS^1⋅S^2, with A=8.6×10−4eV.
((a)) Give the dimension of the joint spin space and its decomposition into coupled multiplets.
((b)) S^1⋅S^2 is a scalar (rank-0 tensor) operator, so it is constant on each multiplet. Compute the energy H^′ gives the triplet (s=1) and the singlet (s=0), and the energy splitting between them.