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=====The $S^{2}$ Operator (15 minutes)=====

  * This lecture is best introduced with the [[swbq:qmsw:spswssquaredmatrix|Computing the $S^{2}$ Operator for the Spin-$\frac{1}{2}$ System]] activity.
  * The $S_{x}$, $S_{y}$, and $S_{z}$ operators for the spin-1 system can then be written on the board.  From here, have a third of the room calculate $S_{x}^{2}$, a third calculate $S_{y}^{2}$, and the final third calculate $S_{z}^{2}$.  Gather answers from each side and write the answers under the non-squared operators.
  * Finally, add all of the squared operators for the spin-1 system to find $S^{2}$.  It can be shown that $S^{2}$ for the spin-1 system is $2\hbar^{2}$ multiplied by the identity matrix.
  * When $2\hbar^{2}$ is compared to the expression $l(l+1)\hbar^{2}$, we can find that $l=1$, indicating that the system is a spin-1 system as we knew before.

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