## Working with Representations on the Ring

• group Wavefunctions on a Quantum Ring

group Small Group Activity

30 min.

##### Wavefunctions on a Quantum Ring
Central Forces 2023 (2 years)
• group Energy and Angular Momentum for a Quantum Particle on a Ring

group Small Group Activity

30 min.

##### Energy and Angular Momentum for a Quantum Particle on a Ring

Quantum Ring Sequence

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.
• group Working with Representations on the Ring

group Small Group Activity

30 min.

##### Working with Representations on the Ring
Central Forces 2023 (3 years)
• group Outer Product of a Vector on Itself

group Small Group Activity

30 min.

##### Outer Product of a Vector on Itself
Quantum Fundamentals 2022 (2 years)

Students compute the outer product of a vector on itself to product a projection operator. Students discover that projection operators are idempotent (square to themselves) and that a complete set of outer products of an orthonormal basis is the identity (a completeness relation).
• face Quantum Reference Sheet

face Lecture

5 min.

##### Quantum Reference Sheet
Central Forces 2023 (6 years)
• assignment The puddle

assignment Homework

##### The puddle
differentials Static Fields 2022 (4 years) The depth of a puddle in millimeters is given by $h=\frac{1}{10} \bigl(1+\sin(\pi xy)\bigr)$ Your path through the puddle is given by $x=3t \qquad y=4t$ and your current position is $x=3$, $y=4$, with $x$ and $y$ also in millimeters, and $t$ in seconds.
1. At your current position, how fast is the depth of water through which you are walking changing per unit time?
2. At your current position, how fast is the depth of water through which you are walking changing per unit distance?
3. FOOD FOR THOUGHT (optional)
There is a walkway over the puddle at $x=10$. At your current position, how fast is the depth of water through which you are walking changing per unit distance towards the walkway.
• assignment Flux through a Paraboloid

assignment Homework

##### Flux through a Paraboloid
Static Fields 2022 (5 years)

Find the upward pointing flux of the electric field $\vec E =E_0\, z\, \hat z$ through the part of the surface $z=-3 s^2 +12$ (cylindrical coordinates) that sits above the $(x, y)$--plane.

• group Applying the equipartition theorem

group Small Group Activity

30 min.

##### Applying the equipartition theorem
Contemporary Challenges 2022 (4 years)

Students count the quadratic degrees of freedom of a few toy molecules to predict their internal energy at temperature $T$.
• assignment Spin Fermi Estimate

assignment Homework

##### Spin Fermi Estimate
Quantum Fundamentals 2022 The following two problems ask you to make Fermi estimates. In a good Fermi estimate, you start from basic scientific facts you already know or quantities that you can reasonably estimate based on your life experiences and then reason your way to estimate a quantity that you would not be able guess. You may look up useful conversion factors or constants. Use words, pictures, and equations to explain your reasoning:
1. Imagine that you send a pea-sized bead of silver through a Stern-Gerlach device oriented to measure the z-component of intrinsic spin. Estimate the total z-component of the intrinsic spin of the ball you would measure in the HIGHLY improbable case that every atom is spin up.
2. Protons, neutrons, and electrons are all spin-1/2 particles. Give a (very crude) order of magnitude estimate of the number of these particles in your body.

assignment Homework

Consider the fields at a point $\vec{r}$ due to a point charge located at $\vec{r}'$.

1. Write down an expression for the electrostatic potential $V(\vec{r})$ at a point $\vec{r}$ due to a point charge located at $\vec{r}'$. (There is nothing to calculate here.)
2. Write down an expression for the electric field $\vec{E}(\vec{r})$ at a point $\vec{r}$ due to a point charge located at $\vec{r}'$. (There is nothing to calculate here.)
3. Working in rectangular coordinates, compute the gradient of $V$.
4. Write several sentences comparing your answers to the last two questions.

• Central Forces 2023 (3 years)

The following are 3 different representations for the $\textbf{same}$ state on a quantum ring for $r_0=1$ $$\left|{\Phi_a}\right\rangle = i\sqrt{\frac{ 2}{12}}\left|{3}\right\rangle - \sqrt{\frac{ 1}{12}}\left|{1}\right\rangle +\sqrt{\frac{ 3}{12}}e^{i\frac{\pi}{4}}\left|{0}\right\rangle -i\sqrt{\frac{ 2}{ 12}}\left|{-1}\right\rangle +\sqrt{\frac{ 4}{12}}\left|{-3}\right\rangle$$ $$\left| \Phi_b\right\rangle \doteq \left( \begin{matrix} \vdots \\ i\sqrt{\frac{ 2}{12}}\\ 0 \\ -\sqrt{\frac{ 1}{12}} \\ \sqrt{\frac{ 3}{12}}e^{i\frac{\pi}{4}} \\ -i\sqrt{\frac{ 2}{12}}\\ 0 \\ \sqrt{\frac{4}{12} }\\ \vdots \end{matrix}\right) \begin{matrix} \leftarrow m=0 \end{matrix}$$ $$\Phi_c(\phi) \doteq \sqrt{\frac{1}{24 \pi}} \left( i\sqrt{2}e^{i 3 \phi} -e^{i\phi} +\sqrt{3}e^{i\frac{\pi}{4}} -i \sqrt{2} e^{-i\phi} + \sqrt{4}e^{-i 3 \phi} \right)$$

1. With each representation of the state given above, explicitly calculate the probability that $L_z=-1\hbar$. Then, calculate all other non-zero probabilities for values of $L_z$ with a method/representation of your choice.
2. Explain how you could be sure you calculated all of the non-zero probabilities.
3. If you measured the $z$-component of angular momentum to be $3\hbar$, what would the state of the particle be immediately after the measurement is made?
4. With each representation of the state given above, explicitly calculate the probability that $E=\frac{9}{2}\frac{\hbar^2}{I}$. Then, calculate all other non-zero probabilities for values of $E$ with a method of your choice.
5. If you measured the energy of the state to be $\frac{9}{2}\frac{\hbar^2}{I}$, what would the state of the particle be immediately after the measurement is made?