Activities
Eigenvalues and EigenvectorsEach 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:
- Find the eigenvalues.
- Find the (unnormalized) eigenvectors.
- Describe what this transformation does.
- 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 doesStudent 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.
Problem
- Find the eigenvalues and normalized eigenvectors of the Pauli matrices \(\sigma_x\), \(\sigma_y\), and \(\sigma_z\) (see the Spins Reference Sheet posted on the course website).
The operator \(\hat{S}_x\) for spin-1 (in the \(z\)-basis) may be written as: \[\hat{S}_x=\frac{\hbar}{\sqrt{2}} \begin{pmatrix} 0&1&0\\ 1&0&1 \\ 0&1&0 \\ \end{pmatrix} \]
- Find the eigenvalues of this matrix.
- Find the (un-normalized) eigenvectors of this matrix. Write the eigenvectors as both matrices and kets.
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.
Problem
- Let \[|\alpha\rangle \doteq \frac{1}{\sqrt{2}} \begin{pmatrix} 1\\ 1 \end{pmatrix} \qquad \rm{and} \qquad |\beta\rangle \doteq \frac{1}{\sqrt{2}} \begin{pmatrix} 1\\ -1 \end{pmatrix}\] Show that \(\left|{\alpha}\right\rangle \) and \(\left|{\beta}\right\rangle \) are orthonormal. (If a pair of vectors is orthonormal, that suggests that they might make a good basis.)
- Consider the matrix \[C\doteq \begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix} \] Show that the vectors \(|\alpha\rangle\) and \(|\beta\rangle\) are eigenvectors of C and find the eigenvalues. (Note that showing something is an eigenvector of an operator is far easier than finding the eigenvectors if you don't know them!)
- A operator is always represented by a diagonal matrix if it is written in terms of the basis of its own eigenvectors. What does this mean? Find the matrix elements for a new matrix \(E\) that corresponds to \(C\) expanded in the basis of its eigenvectors, i.e. calculate \(\langle\alpha|C|\alpha\rangle\), \(\langle\alpha|C|\beta\rangle\), \(\langle\beta|C|\alpha\rangle\) and \(\langle\beta|C|\beta\rangle\) and arrange them into a sensible matrix \(E\). Explain why you arranged the matrix elements in the order that you did.
- Find the determinants of \(C\) and \(E\). How do these determinants compare to the eigenvalues of these matrices?
Problem
Consider the arbitrary Pauli matrix \(\sigma_n=\hat n\cdot\vec \sigma\) where \(\hat n\) is the unit vector pointing in an arbitrary direction.
- Find the eigenvalues and normalized eigenvectors for \(\sigma_n\). The answer is: \[ \begin{pmatrix} \cos\frac{\theta}{2}e^{-i\phi/2}\\{} \sin\frac{\theta}{2}e^{i\phi/2}\\ \end{pmatrix} \begin{pmatrix} -\sin\frac{\theta}{2}e^{-i\phi/2}\\{} \cos\frac{\theta}{2}e^{i\phi/2}\\ \end{pmatrix} \] It is not sufficient to show that this answer is correct by plugging into the eigenvalue equation. Rather, you should do all the steps of finding the eigenvalues and eigenvectors as if you don't know the answer. Hint: \(\sin\theta=\sqrt{1-\cos^2\theta}\).
- Show that the eigenvectors from part (a) above are orthogonal.
- Simplify your results from part (a) above by considering the three separate special cases: \(\hat n=\hat\imath\), \(\hat n=\hat\jmath\), \(\hat n=\hat k\). In this way, find the eigenvectors and eigenvalues of \(\sigma_x\), \(\sigma_y\), and \(\sigma_z\).