## Activity: Spherical Harmonics

Central Forces 2023 (3 years)

Spherical Harmonics:

$\boxed{\begin{array}{lll} \ell& m & \quad\quad\quad\; Y_\ell^m(\theta,\phi) \\[.35cm] \hline \\[.03cm] 0 & 0 & \quad\;\; Y_0^0=\sqrt{\frac{1}{4\pi}} \\[.35cm] 1 & 0 & \quad\;\; Y_1^0=\sqrt{\frac{3}{4\pi}}\cos\theta \\[.35cm] & \pm1 & \quad Y_1^{\pm1}=\mp\sqrt{\frac{3}{8\pi}}\sin\theta e^{\pm i\phi} \\[.35cm] 2 & 0 & \quad\;\;Y_2^0=\sqrt{\frac{5}{16\pi}}\left(3\cos^2\theta-1 \right) \\[.35cm] & \pm1 & \quad Y_2^{\pm1}=\mp\sqrt{\frac{15}{8\pi}}\sin\theta\cos \theta e^{\pm i\phi} \\[.35cm] & \pm2 & \quad Y_2^{\pm2}=\sqrt{\frac{15}{32\pi}}\sin^2\theta e^{\pm2i\phi} \\[.35cm] 3 & 0 & \quad\;\;Y_3^0=\sqrt{\frac{7}{16\pi}}\left(5\cos^3\theta-3 \cos\theta\right) \\[.35cm] & \pm1 & \quad Y_3^{\pm1}=\mp\sqrt{\frac{21}{64\pi}}\sin\theta \left(5\cos^2\theta-1\right)e^{\pm i\phi} \\[.35cm] & \pm2 & \quad Y_3^{\pm2}=\sqrt{\frac{105}{32\pi}} \sin^2\theta\cos\theta e^{\pm2i\phi} \\[.35cm] & \pm3 & \quad Y_3^{\pm3}=\sqrt{\frac{35}{64\pi}}\sin^3\theta e^{\pm3i\phi} \\[.001cm] \end{array}}$

• face Thermal radiation and Planck distribution

face Lecture

120 min.

##### Thermal radiation and Planck distribution
Thermal and Statistical Physics 2020

These notes from the fourth week of Thermal and Statistical Physics cover blackbody radiation and the Planck distribution. They include a number of small group activities.
• group Name the experiment

group Small Group Activity

30 min.

##### Name the experiment
Energy and Entropy 2021 (3 years)

Student groups design an experiment that measures an assigned partial derivative. In a compare-and-contrast wrap-up, groups report on how they would measure their derivatives.
• group Name the experiment (changing entropy)

group Small Group Activity

30 min.

##### Name the experiment (changing entropy)
Energy and Entropy 2021 (2 years)

Students are placed into small groups and asked to create an experimental setup they can use to measure the partial derivative they are given, in which entropy changes.
• group Finding if $S_{x}, \; S_{y}, \; and \; S_{z}$ Commute

group Small Group Activity

30 min.

##### Finding if $S_{x}, \; S_{y}, \; and \; S_{z}$ Commute
Quantum Fundamentals 2022 (3 years)
• group Expectation Value and Uncertainty for the Difference of Dice

group Small Group Activity

60 min.

##### Expectation Value and Uncertainty for the Difference of Dice
Quantum Fundamentals 2022 (3 years)
• group Covariation in Thermal Systems

group Small Group Activity

30 min.

##### Covariation in Thermal Systems

Students consider how changing the volume of a system changes the internal energy of the system. Students use plastic graph models to explore these functions.
• assignment Pressure and entropy of a degenerate Fermi gas

assignment Homework

##### Pressure and entropy of a degenerate Fermi gas
Fermi gas Pressure Entropy Thermal and Statistical Physics 2020
1. Show that a Fermi electron gas in the ground state exerts a pressure \begin{align} p = \frac{\left(3\pi^2\right)^{\frac23}}{5} \frac{\hbar^2}{m}\left(\frac{N}{V}\right)^{\frac53} \end{align} In a uniform decrease of the volume of a cube every orbital has its energy raised: The energy of each orbital is proportional to $\frac1{L^2}$ or to $\frac1{V^{\frac23}}$.

2. Find an expression for the entropy of a Fermi electron gas in the region $kT\ll \varepsilon_F$. Notice that $S\rightarrow 0$ as $T\rightarrow 0$.

• face Angular Momentum Commutation Relations: Lecture

face Lecture

10 min.

##### Angular Momentum Commutation Relations: Lecture
Central Forces 2023 (3 years)
• group DELETE Navigating a Hill

group Small Group Activity

30 min.

##### DELETE Navigating a Hill
Static Fields 2022 (5 years) In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field.
• assignment Photon carnot engine

assignment Homework

##### Photon carnot engine
Carnot engine Work Energy Entropy Thermal and Statistical Physics 2020

In our week on radiation, we saw that the Helmholtz free energy of a box of radiation at temperature $T$ is \begin{align} F &= -8\pi \frac{V(kT)^4}{h^3c^3}\frac{\pi^4}{45} \end{align} From this we also found the internal energy and entropy \begin{align} U &= 24\pi \frac{(kT)^4}{h^3c^3}\frac{\pi^4}{45} V \\ S &= 32\pi kV\left(\frac{kT}{hc}\right)^3 \frac{\pi^4}{45} \end{align} Given these results, let us consider a Carnot engine that uses an empty metalic piston (i.e. a photon gas).

1. Given $T_H$ and $T_C$, as well as $V_1$ and $V_2$ (the two volumes at $T_H$), determine $V_3$ and $V_4$ (the two volumes at $T_C$).

2. What is the heat $Q_H$ taken up and the work done by the gas during the first isothermal expansion? Are they equal to each other, as for the ideal gas?

3. Does the work done on the two isentropic stages cancel each other, as for the ideal gas?

4. Calculate the total work done by the gas during one cycle. Compare it with the heat taken up at $T_H$ and show that the energy conversion efficiency is the Carnot efficiency.

Learning Outcomes