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)}$

Please help it's bugging me.

  • $\begingroup$ This is bad notation as you are not indicating unit vectors... $\endgroup$ – ZeroTheHero Mar 27 at 15:33
  • $\begingroup$ @zerothehero pardon me sir but I typed it in a haste on my android app. $\endgroup$ – user256265 Mar 27 at 15:35

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.

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