• Eigenvalues and eigenstates
• Measurements of energy and angular momentum for hydrogen atom
• Quantum probabilities
• Superposition of states
• Quantum calculations in multiple representations

Estimated Time: 30 minutes

Students are asked to find eigenvalues, probabilities, and expectation values for $H$, $L^2$, and $L_z$ for a superposition of $\vert nlm \rangle$ 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

• Eigenstates & Eigenvalues
• The energy and angular momentum eigenstates and eigenvalues of a hydrogen atom
• Calculating probabilities using Dirac “bra-ket”, matrix and wavefunction notation
• Familiarity with time evolution

This activity works well when sequenced with similar activities for a particle confined to a ring and a particle confined to a sphere.

• Tabletop Whiteboard with markers

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).

• $P_{L_z=m\hbar}$; $\langle L_z\rangle$
• $P_{L^2=\ell(\ell+1)\hbar^2}$; $\langle L^2\rangle$
• $P_{E=-13.6eV/n^2}$; $\langle E\rangle$

Students are then asked to 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. It may be necessary to walk through one calculation to remind them of the method they have used before to generate these matrices.

• Students can often find probabilities by inspection at this point, so it is sometimes helpful to ask them to write out explicitly what they did to lead into the discussion of the summation limits.
• In writing matrices, this is the often the first time that they are only writing a matrix for a subset of the space, so questions about basis, degeneracy, and order crop up.

Probabilities and Expectation Values:

• Summarize explicitly how to calculate the probability of degenerate states by summing the probabilities.
  • It is a good idea to do the wrap-up after each small whiteboard question to help to keep the whole class together.
  • Many students want to avoid explicitly putting the summation in, so this is a good time to talk explicitly about the summation limits for each case. For example, in finding the probability for measuring $L_z$ to be $-1\hbar$, students want to simply write:$$P_{L_z=-\hbar} = |\langle n, \ell, -1| \Psi \rangle |^2$$instead of

$$P_{L_z=-\hbar} = \sum^{\infty}_{n=\vert m\vert+1} \sum^{n-1}_{\ell=\vert m\vert} |\langle n, \ell, m| \Psi \rangle |^2= \sum^{\infty}_{n=2} \sum^{2}_{\ell=1} |\langle n, \ell, -1| \Psi \rangle |^2$$

• This is a good time to also address the difference between the sum of squares and the square of sums, which is an ongoing issue for many students.

Matrix notation:

• If the students use small whiteboards to write $\hat{H}$, $\hat{L}^2$, and $\hat{L}_z$ in matrix notation for this state, you can often find that people have used different ordering systems and by using several different examples, you can highlight several issues:
  • the arbitrariness of the order when there is degeneracy,
  • importance of being consistent with order between operators and vectors,
  • what the “typical” order is for hydrogen states, and
  • matrix notation is not as unwieldy as it seems if you have a small enough subspace.
• You can also highlight the implicit basis ($\vert n\ell m\rangle$) and reiterate the fact that all three matrices are diagonal in this basis (i.e. they share eigenstates).

This is also a good time to talk about the different ways of finding expectation values and when each is appropriate.

• This works well when used in conjunction with the Quantum Ring Sequence.

• You can also use the same state to discuss time dependence again in terms of the Hydrogen atom.

• A good follow-up activity is Probability of Finding an Electron Inside the Bohr Radius, which requires them to think about probability for continuous functions and to do a 3D probability.