Addition of two spin 1/2 operators along an arbitrary direction
I'm trying to figure out how the sum of two spin 1/2 operators along an arbitrary direction would work.
These operators are of the form \begin{equation} \textbf{S}_j \cdot \hat{\textbf{n}}_j = \frac{\hbar}{2} \begin{pmatrix} \cos (\theta_j) & e^{-i \varphi_j} \sin (\theta_j)\ e^{i \varphi_j} \sin (\theta_j) & -\cos (\theta_j) \end{pmatrix}_j \end{equation} where $j = 1, 2$, $\textbf{S}_j = \frac{\hbar}{2} \boldsymbol{\sigma}_j$, and $\hat{\textbf{n}}_j = \sin (\theta_j) \cos (\varphi_j) \hat{\textbf{i}} + \sin (\theta_j) \sin (\varphi_j) \hat{\textbf{j}} + \cos (\theta_j) \hat{\textbf{k}}$. The eigenvalues of these operators are $\pm \frac{\hbar}{2}$ which I'll identify with $\pm$ respectively. Then, the eigenvectors of these eigenvalues are \begin{align} |\textbf{S}_j \cdot \textbf{n}_j, + \rangle &= \cos \Big( \frac{\theta_j}{2} \Big) |S_z, + \rangle^{(j)} + e^{i \varphi_j} \sin \Big( \frac{\theta_j}{2} \Big) |S_z, - \rangle^{(j)}\ |\textbf{S}_j \cdot \textbf{n}_j, - \rangle &= \sin \Big( \frac{\theta_j}{2} \Big) |S_z, + \rangle^{(j)} - e^{i \varphi_j} \cos \Big( \frac{\theta_j}{2} \Big) |S_z, - \rangle^{(j)} \end{align} These are also eigenvectors of $S^2$, both of them with eigenvalue $\frac{3 \hbar^2}{4}$. Let $\textbf{S} \cdot \hat{\textbf{n}} = \textbf{S}_1 \cdot \hat{\textbf{n}}_1 + \textbf{S}_2 \cdot \hat{\textbf{n}}_2$. 281