Main Page/CG coefficients of 3 spin half particles

Problem

(a) Find the total spin of a syatem of three spin ${\frac {1}{2}}$ particles and derive the corresponding Clebsch-Gordan Coefficients.

(b) Consider a system of three nonidentical spin ${\frac {1}{2}}$ particles whose Hamiltonian is given by ${\hat {H}}=-\epsilon _{0}({\vec {S}}_{1}.{\vec {S}}_{3}+{\vec {S}}_{2}.{\vec {S}}_{3})/\hbar ^{2}$ . Find the system's energy levels and their degeneracies.

Solution:-

(a) To add $j_{1}={\frac {1}{2}}$ , $j_{2}={\frac {1}{2}}$ and $j_{3}={\frac {1}{2}}$ , we begin by coupling $j_{1}$ and $j_{2}$ to form $j_{12}=j_{1}+j_{2}$ where $|j_{1}-j_{2}|\leq j_{12}\leq |j_{1}+j_{2}|$ , hence $j_{12}=0,1$ . Then we add j $_{12}$ and $j_{3}$ ; this leads to $|j_{12}-j_{3}|\leq j\leq |j_{12}+j_{3}|$

We are going to denote the joint eigenstates of ${\hat {J_{1}}}^{2}$ , ${\hat {J_{2}}}^{2}$ , ${\hat {J_{3}}}^{2}$ , ${\hat {J_{12}}}^{2}$ , ${\hat {J}}^{2}$ and $J_{z}$ by $|j_{12},j,m>$ and the joint eigenstates of ${\hat {J_{1}}}^{2}$ , ${\hat {J_{2}}}^{2}$ , ${\hat {J_{3}}}^{2}$ , ${\hat {J_{1_{z}}}}^{2}$ , ${\hat {J_{2_{z}}}}^{2}$ and ${\hat {J_{3_{z}}}}^{2}$ by $|j_{1},J_{2},j_{3};m_{1},m_{2},m_{3}>$ ; since $j_{1}=j_{2}=j_{3}={\frac {1}{2}}$ and $m_{1}=\pm {\frac {1}{2}}$ , $m_{2}=\pm {\frac {1}{2}}$ , $m_{3}=\pm {\frac {1}{2}}$ ,we will be using throughout this problem the lighter notation $|j_{1},j_{2},j_{3};\pm ,\pm ,\pm >$ to abbreviate $|{\frac {1}{2}},{\frac {1}{2}},{\frac {1}{2}};\pm {\frac {1}{2}},\pm {\frac {1}{2}},\pm {\frac {1}{2}}>$