spins_unit_operators_and_measurements.ppt Pages 34-37

• Begin by stating that if two operators commute, they share eigenvectors.
• Write down on the board that the commutator is defined as

$$\left[S_{x},S_{y}\right]=S_{x}S_{y}-S_{y}S_{x} \; \; .$$

• This is a good time to perform the “Finding if $S_{x}$, $S_{y}$, and $S_{z}$ Commute” activity. Write down the solutions onto the board.
• Ask the class if they notice any pattern in the answers to the commutators. Point out the cycle that occurs between the x, y, and z letters that takes the form

$$\left[S_{x},S_{y}\right]=i\hbar S_{z} \; \; , $$

$$\left[S_{y},S_{z}\right]=i\hbar S_{x} \; \; , $$

$$\left[S_{z},S_{x}\right]=i\hbar S_{y} \; \; . $$

Note to the class this is commonly just written as

$$\left[S_{x},S_{y}\right]=i\hbar S_{z} \; \; (\text{+ cyclic}) \; \; . $$

Be sure to tell the class that these relations are called the “Angular Momentum Commutation Relations” and that they show up very frequently. State that these relations will hold true for the angular momentum in all cases.

• Now, compute $\left[\hat{S_{z}},\hat{H}\right]$ and show that the two commute. This indicates that the z-basis is a good choice for the Hamiltonian because it shares eigenvectors with $\hat{S_{z}}$.
• Be sure to tell the class that the Hamiltonian will not commute with $S_{x}$ or $S_{y}$. So, the x and y-bases are a bad choice for the Hamiltonian in this case.
• Also show that $S^{2}$ commutes with $\hat{S_{x}}$, $\hat{S_{y}}$, and $\hat{S_{z}}$. Recall that since $S^2$ had the form of the identity matrix, everything vector was an eigenvector. Thus, the fact that $S^{2}$ shares eigenvectors with the x, y, and z spin operators is expected.