A two-nucleon system is in a spin-singlet state. The spins are represented by Pauli matrices $\boldsymbol\sigma_1$ and $\boldsymbol\sigma_2$. Find $\boldsymbol\sigma_1\cdot\boldsymbol\sigma_2$ as an integer.
Write $\mathbf S_i=\hbar\boldsymbol\sigma_i/2$ and $\mathbf S=\mathbf S_1+\mathbf S_2$. Then $S^2=S_1^2+S_2^2+2\mathbf S_1\cdot\mathbf S_2$. In the singlet, $S^2=0$, while each spin-half particle has $S_i^2=3\hbar^2/4$. Thus $\mathbf S_1\cdot\mathbf S_2=-3\hbar^2/4$ and $\boldsymbol\sigma_1\cdot\boldsymbol\sigma_2=4\mathbf S_1\cdot\mathbf S_2/\hbar^2=-3$.
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