In quantum mechanics, it turns out that the overall phase for a state does not have any physical significance. Therefore, you will need to become quick at rearranging the phase of various states. For each of the vectors listed below, rewrite the vector as an overall complex phase times a new vector whose first component is real and positive. \[\left|D\right\rangle\doteq \begin{pmatrix} 7e^{i\frac{\pi}{6}}\\ 3e^{i\frac{\pi}{2}}\\ -1\\ \end{pmatrix}\\ \left|E\right\rangle\doteq \begin{pmatrix} i\\ 4\\ \end{pmatrix}\\ \left|F\right\rangle\doteq \begin{pmatrix} 2+2i\\ 3-4i\\ \end{pmatrix} \]

Students use completeness relations to write a matrix element of a spin component in a different basis.

In this small group activity, students solve for the time dependence of two quantum spin 1/2 particles under the influence of a Hamiltonian. Students determine, given a Hamiltonian, which states are stationary and under what circumstances measurement probabilities do change with time.

This activity acts as a reintroduction to doing quantum calculations while also introducing the matrix representation on the ring, allowing students to discover how to index and form a column vector representing the given quantum state. In addition, this activity introduces degenerate measurements on the quantum ring and examines the state after measuring both degenerate and non-degenerate eigenvalues for the state.

Students consider the relation (1) between the angular momentum and magnetic moment for a current loop and (2) the force on a magnetic moment in an inhomogeneous magnetic field. Students make a (classical) prediction of the outcome of a Stern-Gerlach experiment.

Students work in small groups to use completeness relations to change the basis of quantum states.

Students use their arms to act out two spin-1/2 quantum states and their inner product.

Students use Mathematica to visualize the probability density distribution for the hydrogen atom orbitals with the option to vary the values of \(n\), \(\ell\), and \(m\).

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.

Eigenvalues and Eigenvectors

Each group will be assigned one of the following matrices.

\[ A_1\doteq \begin{pmatrix} 0&-1\\ 1&0\\ \end{pmatrix} \hspace{2em} A_2\doteq \begin{pmatrix} 0&1\\ 1&0\\ \end{pmatrix} \hspace{2em} A_3\doteq \begin{pmatrix} -1&0\\ 0&-1\\ \end{pmatrix} \]
\[ A_4\doteq \begin{pmatrix} a&0\\ 0&d\\ \end{pmatrix} \hspace{2em} A_5\doteq \begin{pmatrix} 3&-i\\ i&3\\ \end{pmatrix} \hspace{2em} A_6\doteq \begin{pmatrix} 0&0\\ 0&1\\ \end{pmatrix} \hspace{2em} A_7\doteq \begin{pmatrix} 1&2\\ 1&2\\ \end{pmatrix} \]
\[ A_8\doteq \begin{pmatrix} -1&0&0\\ 0&-1&0\\ 0&0&-1\\ \end{pmatrix} \hspace{2em} A_9\doteq \begin{pmatrix} -1&0&0\\ 0&-1&0\\ 0&0&1\\ \end{pmatrix} \]
\[ S_x\doteq \frac{\hbar}{2}\begin{pmatrix} 0&1\\ 1&0\\ \end{pmatrix} \hspace{2em} S_y\doteq \frac{\hbar}{2}\begin{pmatrix} 0&-i\\ i&0\\ \end{pmatrix} \hspace{2em} S_z\doteq \frac{\hbar}{2}\begin{pmatrix} 1&0\\ 0&-1\\ \end{pmatrix} \]

For your matrix:

1. Find the eigenvalues.
2. Find the (unnormalized) eigenvectors.
3. Describe what this transformation does.
4. Normalize your eigenstates.

If you finish early, try another matrix with a different structure, i.e. real vs. complex entries, diagonal vs. non-diagonal, \(2\times 2\) vs. \(3\times 3\), with vs. without explicit dimensions.

Instructor's Guide

Main Ideas

This is a small group activity for groups of 3-4. The students will be given one of 10 matrices. The students are then instructed to find the eigenvectors and eigenvalues for this matrix and record their calculations on their medium-sized whiteboards. In the class discussion that follows students report their finding and compare and contrast the properties of the eigenvalues and eigenvectors they find. Two topics that should specifically discussed are the case of repeated eigenvalues (degeneracy) and complex eigenvectors, e.g., in the case of some pure rotations, special properties of the eigenvectors and eigenvalues of hermitian matrices, common eigenvectors of commuting operators.

Students' Task

Introduction

Give a mini-lecture on how to calculate eigenvalues and eigenvectors. It is often easiest to do this with an example. We like to use the matrix \[A_7\doteq\begin{pmatrix}1&2\cr 9&4\cr\end{pmatrix}\] from the https://paradigms.oregonstate.edu/activities/2179 https://paradigms.oregonstate.edu/activities/2179 Finding Eigenvectors and Eigenvalues since the students have already seen this matrix and know what it's eigenvectors are. Then every group is given a handout, assigned a matrix, and then asked to: - Find the eigenvalues - Find the (unnormalized) eigenvectors - Normalize the eigenvectors - Describe what this transformation does

Student Conversations

• Typically, students can find the eigenvalues without too much problem. Eigenvectors are a different story. To find the eigenvectors, they will have two equations with two unknowns. They expect to be able to find a unique solution. But, since any scalar multiple of an eigenvector is also an eigenvector, their two equations will be redundant. Typically, they must choose any convenient value for one of the components (e.g. \(x=1\)) and solve for the other one. Later, they can use this scale freedom to normalize their vector.
• The examples in this activity were chosen to include many of the special cases that can trip students up. A common example is when the two equations for the eigenvector amount to something like \(x=x\) and \(y=-y\). For the first equation, they may need help to realize that \(x=\) “anything” is the solution. And for the second equation, sadly, many students need to be helped to the realization that the only solution is \(y=0\).

Wrap-up

The majority of the this activity is in the wrap-up conversation.

The [[whitepapers:narratives:eigenvectorslong|Eigenvalues and Eigenvectors Narrative]] provides a detailed narrative interpretation of this activity, focusing on the wrap-up conversation.

• Complex eigenvectors: connect to discussion of rotations in the Linear Transformations activity where there did not appear to be any vectors that stayed the same.
• Degeneracy: Define degeneracy as the case when there are repeated eigenvalues. Make sure the students see that, in the case of degeneracy, an entire subspace of vectors are all eigenvectors.
• Diagonal Matrices: Discuss that diagonal matrices are trivial. Their eigenvalues are just their diagonal elements and their eigenvectors are just the standard basis.
• Matrices with dimensions: Students should see from these examples that when you multiply a transformation by a real scalar, its eigenvalues are multiplied by that scalar and its eigenvectors are unchanges. If the scalar has dimensions (e.g. \(\hbar/2\)), then the eigenvalues have the same dimensions.

Students compute probabilities and averages given a probability density in one dimension. This activity serves as a soft introduction to the particle in a box, introducing all the concepts that are needed.

Students compute inner products to expand a wave function in a sinusoidal basis set. This activity introduces the inner product for wave functions, and the idea of approximating a wave function using a finite set of basis functions.

Students find matrix elements of the position operator \(\hat x\) in a sinusoidal basis. This allows them to express this operator as a matrix, which they can then numerically diagonalize and visualize the eigenfunctions.

Students implement a finite-difference approximation for the kinetic energy operator as a matrix, and then use numpy to solve for eigenvalues and eigenstates, which they visualize.

In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.

• Work as area under curve in a \(pV\) plot
• Heat transfer as area under a curve in a \(TS\) plot
• Reminder that internal energy is a state function
• Reminder of First Law

Students consider the dimensions of spin-state kets and position-basis kets.

In this small group activity, students work out the steady state temperature of an object absorbing and emitting blackbody radiation.

Students practice using inner products to find the components of the cartesian basis vectors in the polar basis and vice versa. Then, students use a completeness relation to change bases or cartesian/polar bases and for different spin bases.

Students take the inner product of vectors that lie on the spacetime axis to show that they are orthogonal. To do the inner product, students much use the Minkowski metric.

In this small group activity, students multiply a general 3x3 matrix with standard basis row/column vectors to pick out individual matrix elements. Students generate the expressions for the matrix elements in bra/ket notation.

A short improvisational role-playing skit based on the Star Trek series in which students explore the definition and notation for position vectors, the importance of choosing an origin, and the geometric nature of the distance formula. \[\vert\vec{r}-\vec{r}^\prime\vert=\sqrt{(x-x^\prime)^2+(y-y^\prime)^2-(z-z^\prime)^2}\]

In this activity, students apply the Stefan-Boltzmann equation and the principle of energy balance in steady state to find the steady state temperature of a black object in near-Earth orbit.

Students explore what linear transformation matrices do to vectors. The whole class discussion compares & contrasts several different types of transformations (rotation, flip, projections, “scrinch”, scale) and how the properties of the matrices (the determinant, symmetries, which vectors are unchanged) are related to these transformations.