# In addition of angular momenta of two half spin particles, when we try to find square of the total momenta where do some components go?

Reference :- Page 189 Griffiths's Introduction To Quantum Mechanics

He supposes the momenta of two particles to be $$S^{(1)}$$ and $$S^{(2)}$$

The Spin Momenta $$S^{(1)}$$ get vector components as $$S^{(1)}= S_x^{(1)} +S_y^{(1)} +S_z^{(1)}$$

same for $$S^{(2)}$$ as well.

Later he says when both act as a single entity the total momenta comes $$S=S^{(1)}+S^{(2)}$$ and when he calculates $$S^{(1)} \cdot S^{(2)}$$ in $$S^2$$

He just writes products of $$S_x^{(1)} \cdot S_x^{(2)}$$ , $$S_y^{(1)} \cdot S_y^{(2)}$$ and $$S_z^{(1)} \cdot S_z^{(2)}$$

What about $$S_x^{(1)} \cdot S_y^{(2)}$$ or $$S_x^{(1)} \cdot S_z^{(2)}$$ or $$S_y^{(1)} \cdot S_x^{(2)}$$

The issue is your notation. You should really have $$\textbf{S}^{(1)}= \hat{\textbf{x}}S_x^{(1)}+\hat{\textbf{y}}S_y^{(1)}+\hat{\textbf{z}}S_z^{(1)}$$ and likewise with your $$\textbf{S}^{(2)}$$. In taking $$\textbf{S}^{(1)}\cdot \textbf{S}^{(2)}$$ the cross-terms do not appear because $$\hat{\textbf{x}}\cdot\hat{\textbf{y}}=0$$ etc.