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.

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.

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

This activity allows students to puzzle through indexing, the from of operators in quantum mechanics, and working with the new quantum numbers on the sphere in an applied context.

For systems of particles, we used the formulas \begin{align} \vec{R}_{cm}&=\frac{1}{M}\left(m_1\vec{r}_1+m_2\vec{r}_2\right) \nonumber\\ \vec{r}&=\vec{r}_2-\vec{r}_1 \label{cm} \end{align} to describe the system of two objects in terms of the center of mass and relative position instead of the positions of each object. After solving for the equations of motion in the center-of-mass coordinates, you may want to transform back to the original coordinate system to examine the motion of each object.

Find the positions of the two objects in terms of the position of the center of mass and the relative position, i.e. solve for: \begin{align} \vec{r}_1&=\\ \vec{r}_2&= \end{align}

Hint: The system of equations (\ref{cm}) is linear, i.e. each variable is to the first power, even though the variables are vectors. In this case, you can use all of the methods you learned for solving systems of equations while keeping the variables vector valued, i.e. you can safely ignore the fact that the \(\vec{r}\)s are vectors while you are doing the algebra as long as you don't divide by a vector.

Review, as much as necessary, how to do matrix addition, multiplication of a matrix by a scalar, matrix multiplication, determinant of a matrix (\(2\times 2\) and \(3\times 3\)), and inverse of a matrix (\(2\times 2\) only). You might find the information at the following links useful.

• Matrix Addition
• Scalar Multiplication
• Matrix Multiplication
• Determinants
• Inverses

There is nothing to turn in, but it will help you get familiar with the linear algebra that we will frequently use in this course.

Overview paragraph here.

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.

The Pauli spin matrices \(\sigma_x\), \(\sigma_y\), and \(\sigma_z\) are defined by: \[\sigma_x= \begin{pmatrix} 0&1\\ 1&0\\ \end{pmatrix} \hspace{2em} \sigma_y= \begin{pmatrix} 0&-i\\ i&0\\ \end{pmatrix} \hspace{2em} \sigma_z= \begin{pmatrix} 1&0\\ 0&-1\\ \end{pmatrix} \] These matrices are related to angular momentum in quantum mechanics.

1. By drawing pictures, convince yourself that the arbitrary unit vector \(\hat n\) can be written as: \[\hat n=\sin\theta\cos\phi\, \hat x +\sin\theta\sin\phi\,\hat y+\cos\theta\,\hat z\] where \(\theta\) and \(\phi\) are the parameters used to describe spherical coordinates.
2. Find the entries of the matrix \(\hat n\cdot\vec \sigma\) where the “matrix-valued-vector” \(\vec \sigma\) is given in terms of the Pauli spin matrices by \[\vec\sigma=\sigma_x\, \hat x + \sigma_y\, \hat y+\sigma_z\, \hat z\] and \(\hat n\) is given in part (a) above.

For this problem, use the vectors \(|a\rangle = 4 |1\rangle - 3 |2\rangle\) and \(|b\rangle = -i |1\rangle + |2\rangle\).

1. Find \(\langle a | b \rangle\) and \(\langle b | a \rangle\). Discuss how these two inner products are related to each other.
2. For \(\hat{Q}\doteq \begin{pmatrix} 2 & i \\ -i & -2 \end{pmatrix} \), calculate \(\langle1|\hat{Q}|2\rangle\), \(\langle2|\hat{Q}|1\rangle\), \(\langle a|\hat{Q}| b \rangle\) and \(\langle b|\hat{Q}|a \rangle\).
3. What kind of mathematical object is \(|a\rangle\langle b|\)? What is the result if you multiply a ket (for example, \(| a\rangle\) or \(|1\rangle\)) by this expression? What if you multiply this expression by a bra?

1. 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.)
2. 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!)
3. 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.
4. Find the determinants of \(C\) and \(E\). How do these determinants compare to the eigenvalues of these matrices?

Students each recall a representation of vectors that they have seen before and record it on an individual whiteboard. The instructor uses these responses to generate a whole class discussion that compares and contrasts the features of the representations. If appropriate for the class, the instructor introduces bra/ket notation as a new, but valuable representation.

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.

Students review using the Arms representation to represent states for discrete quantum systems and connecting the Arms representation to histogram and matrix representation. The student then extend the Arms representation to begin exploring the continuous position basis.

• How to form a state as a column vector in matrix representation.
• How to do probability calculations on all three representations used for quantum systems in PH426.
• How to find probabilities for and the resultant state after measuring degenerate eigenvalues.

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.

Students calculate probabilities for a particle on a ring using three different notations: Dirac bra-ket, matrix, and wave function. After calculating the angular momentum and energy measurement probabilities, students compare their calculation methods for notation.

Students, working in pairs, use their left arms to represent each component in a two-state quantum spin 1/2 system. Reinforces the idea that quantum states are complex valued vectors. Students make connections between Dirac, matrix, and Arms representation.

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.

Students re-represent a state given in Dirac notation in matrix notation

• Divergence of a vector field (at a point) is the flux per unit volume through an infinitesimal box.
• How to predict the sign and relative magnitude of the divergence from graphs of a vector field.
• (Optional) How to calculate the divergence of a vector field with computer algebra.

• A component of the curl of a vector field (at a point) is the circulation per unit area around an infinitesimal loop.
• How to predict the sign and relative magnitude of the curl from graphs of a vector field.
• (Optional) How to calculate the curl of a vector field using computer algebra.

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.