This activity reinforces the strategies students have been practicing on each system by letting them create their own matrix operators and columns on the hydrogen atom and do some calculations with them.
For the state \[ \left|{\Psi}\right\rangle = \sqrt{\frac{7}{10}} |2, 1, 0\rangle + \sqrt{\frac{1}{10}} |3, 2, 1\rangle + i\sqrt{\frac{2}{10}} |3, 1, 1\rangle\]
- Convert the state to matrix notation.
- Find \(\mathcal{P}(L_z=0\hbar)\)
- Find \(\langle E\rangle\)
Students are asked to find eigenvalues, probabilities, and expectation values for \(H\), \(L^2\), and \(L_z\) for a superposition of \(\vert n\ell m \rangle\) states. This can be done on small whiteboards or with the students working in groups on large whiteboards.
Students then work together in small groups to find the matrices that correspond to \(H\), \(L^2\), and \(L_z\) and to redo \(\langle E\rangle\) in matrix notation
This is also a good time to talk about the different ways of finding expectation values and when each is appropriate.