*group*Representations of the Infinite Square Well*group*Small Group Activity120 min.

##### Representations of the Infinite Square Well

Quantum Fundamentals 2022 (3 years)*assignment*Phase 2*assignment*Homework##### Phase 2

quantum mechanics relative phase overall phase measurement probability Quantum Fundamentals 2022 (2 years) Consider the three quantum states: \[\left\vert \psi_1\right\rangle = \frac{4}{5}\left\vert +\right\rangle+ i\frac{3}{5} \left\vert -\right\rangle\] \[\left\vert \psi_2\right\rangle = \frac{4}{5}\left\vert +\right\rangle- i\frac{3}{5} \left\vert -\right\rangle\] \[\left\vert \psi_3\right\rangle = -\frac{4}{5}\left\vert +\right\rangle+ i\frac{3}{5} \left\vert -\right\rangle\]- For each of the \(\left|{\psi_i}\right\rangle \) above, calculate the probabilities of spin component measurements along the \(x\), \(y\), and \(z\)-axes.
*Look For a Pattern (and Generalize):*Use your results from \((a)\) to comment on the importance of the overall phase and of the relative phases of the quantum state vector.

*accessibility_new*Curvilinear Basis Vectors*accessibility_new*Kinesthetic10 min.

##### Curvilinear Basis Vectors

Static Fields 2022 (8 years) Students use their arms to depict (sequentially) the different cylindrical and spherical basis vectors at the location of their shoulder (seen in relation to a specified origin of coordinates: either a set of axes hung from the ceiling of the room or perhaps a piece of furniture or a particular corner of the room).*assignment*Undo Formulas for Reduced Mass (Algebra)*assignment*Homework##### Undo Formulas for Reduced Mass (Algebra)

Central Forces 2023 (2 years) 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 switch from a rectangular coordinate system that is unrelated to the system to coordinates adapted to the center-of-mass. After you have solved the equations of motion in the center-of-mass coordinates, you may want to transform back to the original coordinate system. Find the inverse transformation, 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.*assignment*Linear Quadrupole (w/ series)*assignment*Homework##### Linear Quadrupole (w/ series)

Static Fields 2022 (5 years)Consider a collection of three charges arranged in a line along the \(z\)-axis: charges \(+Q\) at \(z=\pm D\) and charge \(-2Q\) at \(z=0\).

- Find the electrostatic potential at a point \(\vec{r}\) in the \(xy\)-plane at a distance \(s\) from the center of the quadrupole. The formula for the electrostatic potential \(V\) at a point \(\vec{r}\) due to a charge \(Q\) at the point \(\vec{r'}\) is given by: \[ V(\vec{r})=\frac{1}{4\pi\epsilon_0} \frac{Q}{\vert \vec{r}-\vec{r'}\vert} \] Electrostatic potentials satisfy the superposition principle.
- Assume \(s\gg D\). Find the first two non-zero terms of a power series expansion to the electrostatic potential you found in the first part of this problem.

*face*Time Evolution Refresher (Mini-Lecture)*face*Lecture30 min.

##### Time Evolution Refresher (Mini-Lecture)

Central Forces 2023 (3 years) The instructor gives a brief lecture about time dependence of energy eigenstates (e.g. McIntyre, 3.1). Notes for the students are attached.*group*Systems of Equations Compare and Contrast*keyboard*Kinetic energy*keyboard*Computational Activity120 min.

##### Kinetic energy

Computational Physics Lab II 2022finite difference hamiltonian quantum mechanics particle in a box 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.*face*Gibbs entropy approach*face*Lecture120 min.

##### Gibbs entropy approach

Thermal and Statistical Physics 2020Gibbs entropy information theory probability statistical mechanics

These lecture notes for the first week of Thermal and Statistical Physics include a couple of small group activities in which students work with the Gibbs formulation of the entropy.*keyboard*Mean position*keyboard*Computational Activity120 min.

##### Mean position

Computational Physics Lab II 2022probability density particle in a box wave function quantum mechanics

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.-
Quantum Fundamentals 2022 (2 years)
With the Spins simulation set for a spin 1/2 system, measure the probabilities of all the possible spin components for each of the unknown initial states \(\left|{\psi_3}\right\rangle \) and \(\left|{\psi_4}\right\rangle \).
- Use your measured probabilities to find each of the unknown states as a linear superposition of the \(S_z\)-basis states \(\left|{+}\right\rangle \) and \(\left|{-}\right\rangle \).
*Articulate a Process:*Write a set of general instructions that would allow another student in next year's class to find an unknown state from measured probabilities.*Compare Theory with Experiment:*Design an experiment that will allow you to test whether your prediction for each of the unknown states is correct. Describe your experiment here, clearly but succinctly, as if you were writing it up for a paper. Do the experiment and discuss your results.*Make a Conceptual Connection:*In general, can you determine a quantum state with spin-component probability measurements in only two spin-component-directions? Why or why not?