Quantum Mechanics: Concepts and Applications · Problem 7.32

Identical Particles — Nouredine Zettili

Physics — Quantum Mechanics · spin-hamiltonian-energy-levels

Problem

Consider a system of three nonidentical particles, each of spin

s = \frac{1}{2}

, whose Hamiltonian is given by

\hat{H} = \frac{\varepsilon_1}{\hbar^2}(\hat{S}_1 + \hat{S}_3)\cdot\hat{S}_2 + \frac{\varepsilon_2}{\hbar^2}(\hat{S}_{1_z} + \hat{S}_{2_z} + \hat{S}_{3_z})^2,

where

\varepsilon_1

and

\varepsilon_2

are constants having the dimensions of energy. Find the system's energy levels and their degeneracies.