Commutation Relations for Spin OperatorsA commutator of two observables is defined as:
\[[\hat{A},\hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}\]
Determine the results of the following commutators:
- \([\hat{S}_x,\hat{S}_y]\)
- \([\hat{S}_y,\hat{S}_z]\)
- \([\hat{S}_z,\hat{S}_x]\)
- \([\hat{S}_y,\hat{S}_x]\)
- \([\hat{S}_z,\hat{S}_y]\)
- \([\hat{S}_x,\hat{S}_z]\)
Remember that the matrix representation of the spin operators written in the \(S_z\) basis is: \begin{align*} \hat{S}_x \doteq \frac{\hbar}{2}\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \quad \hat{S}_y \doteq \frac{\hbar}{2}\begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} \quad \hat{S}_z \doteq \frac{\hbar}{2}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \end{align*}
Groups should find that none of the quantum operators commute and therefore do not share the same basis for their respective eigenvectors. Because of this, it provides mathematical evidence for many properties that have so far been only observed. Since none of them commute, none of them have the same base, nor can the spin operators be measured simultaneously.
group Small Group Activity
30 min.
group Small Group Activity
30 min.
assignment Homework
accessibility_new Kinesthetic
10 min.
Quantum State Vectors Complex Numbers Spin 1/2 Arms Representation
Students, working in pairs, use their left arms to represent each component in a two-state quantum spin 1/2 system. Reinforces the idea that quantum states are complex valued vectors. Students make connections between Dirac, matrix, and Arms representation.group Small Group Activity
10 min.
assignment Homework
group Small Group Activity
60 min.
group Small Group Activity
10 min.