User:Sansern

where ${\hat {\mathbf {S} }}_{j}=({\hat {S}}_{j}^{x},{\hat {S}}_{j}^{y},{\hat {S}}_{j}^{z})={\frac {\hbar }{2}}({\hat {\sigma }}_{x},{\hat {\sigma }}_{y},{\hat {\sigma }}_{z})$ is the vector operator for spin-1/2 with $({\hat {\sigma }}_{x},{\hat {\sigma }}_{y},{\hat {\sigma }}_{z})$ being the vector of the Pauli matrices.