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.
For the state \[ \left|{\Psi}\right\rangle = \sqrt{\frac{7}{10}} |2, 1, 0\rangle + \sqrt{\frac{1}{10}} |3, 2, 1\rangle + i\sqrt{\frac{2}{10}} |3, 1, 1\rangle\]
Calculate
- \(\mathcal{P}(L_z=\hbar)\)
\(\langle L_z\rangle\)
Then, if you have time, continue with these calculations:
- \(\mathcal{P}(L^2=2\hbar^2)\)
- \(\langle L^2\rangle\)
- \(\mathcal{P}(E=-13.6eV/3^2=-1.51eV)\)
- \(\langle E\rangle\)
- What measurements can be degenerate on the Hydrogen atom?
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 in ket notation. This can be done on small whiteboards or with the students working in groups on large whiteboards.
Write a linear combination of \(\vert nlm\rangle\) states on the board. For example: \[ \Psi = \sqrt{\frac{7}{10}} |2, 1, 0\rangle + \sqrt{\frac{1}{10}} |3, 2, 1\rangle + i\sqrt{\frac{2}{10}} |3, 1, 1\rangle\] (it is a good idea to provide a state that is degenerate in one or more of the quantum numbers).
Then ask the students a series of small whiteboard questions with a short wrap-up after each one that reiterates key points (see wrap-up below).