## Activity: Hydrogen emission

Contemporary Challenges 2022 (5 years)
In this activity students work out energy level transitions in hydrogen that lead to visible light.
• Media
This activity follows the lecture Quantum waves and spectra
Here is an expression for the energy levels of an electron in a particular situation (an electron bound to a single proton): \begin{align} E_n &= \frac{-13.6\text{ eV}}{n^2} & n&=1,2,3,\cdots \end{align}
1. Sketch the energy level diagram
2. Find at least one allowed transition which produces a photon with a visible color
Note:
\begin{align} \lambda &= \frac{2\pi\hbar c}{E_{\text{photon}}}\\ &= \frac{1240\text{ eV}\cdot\text{nm}}{E_{\text{photon}}} \end{align}
• computer Blackbody PhET

computer Computer Simulation

30 min.

##### Blackbody PhET
Contemporary Challenges 2022 (4 years)

Students use a PhET to explore properties of the Planck distribution.
• assignment Heat capacity of vacuum

assignment Homework

##### Heat capacity of vacuum
Heat capacity entropy Thermal and Statistical Physics 2020
1. Solve for the heat capacity of a vacuum, given the above, and assuming that photons represent all the energy present in vacuum.
2. Compare the heat capacity of vacuum at room temperature with the heat capacity of an equal volume of water.
• assignment Diatomic hydrogen

assignment Homework

##### Diatomic hydrogen
rigid rotor hamiltonian angular momentum ground state hydrogen diatomic probability Energy and Entropy 2021 (2 years)

At low temperatures, a diatomic molecule can be well described as a rigid rotor. The Hamiltonian of such a system is simply proportional to the square of the angular momentum \begin{align} H &= \frac{1}{2I}L^2 \end{align} and the energy eigenvalues are \begin{align} E_{\ell m} &= \hbar^2 \frac{\ell(\ell+1)}{2I} \end{align}

1. What is the energy of the ground state and the first and second excited states of the $H_2$ molecule? i.e. the lowest three distinct energy eigenvalues.

2. At room temperature, what is the relative probability of finding a hydrogen molecule in the $\ell=0$ state versus finding it in any one of the $\ell=1$ states?
i.e. what is $P_{\ell=0,m=0}/\left(P_{\ell=1,m=-1} + P_{\ell=1,m=0} + P_{\ell=1,m=1}\right)$

3. At what temperature is the value of this ratio 1?

4. At room temperature, what is the probability of finding a hydrogen molecule in any one of the $\ell=2$ states versus that of finding it in the ground state?
i.e. what is $P_{\ell=0,m=0}/\left(P_{\ell=2,m=-2} + P_{\ell=2,m=-1} + \cdots + P_{\ell=2,m=2}\right)$

• accessibility_new Time Dilation Light Clock Skit

accessibility_new Kinesthetic

5 min.

##### Time Dilation Light Clock Skit

Students act out the classic light clock scenario for deriving time dilation.
• face Phase transformations

face Lecture

120 min.

##### Phase transformations
Thermal and Statistical Physics 2020

These lecture notes from the ninth week of Thermal and Statistical Physics cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.
• group Heat capacity of N$_2$

group Small Group Activity

30 min.

##### Heat capacity of N2
Contemporary Challenges 2022 (4 years)

Students sketch the temperature-dependent heat capacity of molecular nitrogen. They apply the equipartition theorem and compute the temperatures at which degrees of freedom “freeze out.”
• assignment Pressure of thermal radiation

assignment Homework

Thermal radiation Pressure Thermal and Statistical Physics 2020

(modified from K&K 4.6) We discussed in class that \begin{align} p &= -\left(\frac{\partial F}{\partial V}\right)_T \end{align} Use this relationship to show that

1. \begin{align} p &= -\sum_j \langle n_j\rangle\hbar \left(\frac{d\omega_j}{dV}\right), \end{align} where $\langle n_j\rangle$ is the number of photons in the mode $j$;

2. Solve for the relationship between pressure and internal energy.

• assignment Radiation in an empty box

assignment Homework

##### Radiation in an empty box
Thermal physics Radiation Free energy Thermal and Statistical Physics 2020

As discussed in class, we can consider a black body as a large box with a small hole in it. If we treat the large box a metal cube with side length $L$ and metal walls, the frequency of each normal mode will be given by: \begin{align} \omega_{n_xn_yn_z} &= \frac{\pi c}{L}\sqrt{n_x^2 + n_y^2 + n_z^2} \end{align} where each of $n_x$, $n_y$, and $n_z$ will have positive integer values. This simply comes from the fact that a half wavelength must fit in the box. There is an additional quantum number for polarization, which has two possible values, but does not affect the frequency. Note that in this problem I'm using different boundary conditions from what I use in class. It is worth learning to work with either set of quantum numbers. Each normal mode is a harmonic oscillator, with energy eigenstates $E_n = n\hbar\omega$ where we will not include the zero-point energy $\frac12\hbar\omega$, since that energy cannot be extracted from the box. (See the Casimir effect for an example where the zero point energy of photon modes does have an effect.)

Note
This is a slight approximation, as the boundary conditions for light are a bit more complicated. However, for large $n$ values this gives the correct result.

1. Show that the free energy is given by \begin{align} F &= 8\pi \frac{V(kT)^4}{h^3c^3} \int_0^\infty \ln\left(1-e^{-\xi}\right)\xi^2d\xi \\ &= -\frac{8\pi^5}{45} \frac{V(kT)^4}{h^3c^3} \\ &= -\frac{\pi^2}{45} \frac{V(kT)^4}{\hbar^3c^3} \end{align} provided the box is big enough that $\frac{\hbar c}{LkT}\ll 1$. Note that you may end up with a slightly different dimensionless integral that numerically evaluates to the same result, which would be fine. I also do not expect you to solve this definite integral analytically, a numerical confirmation is fine. However, you must manipulate your integral until it is dimensionless and has all the dimensionful quantities removed from it!

2. Show that the entropy of this box full of photons at temperature $T$ is \begin{align} S &= \frac{32\pi^5}{45} k V \left(\frac{kT}{hc}\right)^3 \\ &= \frac{4\pi^2}{45} k V \left(\frac{kT}{\hbar c}\right)^3 \end{align}

3. Show that the internal energy of this box full of photons at temperature $T$ is \begin{align} \frac{U}{V} &= \frac{8\pi^5}{15}\frac{(kT)^4}{h^3c^3} \\ &= \frac{\pi^2}{15}\frac{(kT)^4}{\hbar^3c^3} \end{align}

• face Energy and heat and entropy

face Lecture

30 min.

##### Energy and heat and entropy
Energy and Entropy 2021 (2 years)

This short lecture introduces the ideas required for Ice Calorimetry Lab or Microwave oven Ice Calorimetry Lab.
• 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
• ph315: 8) Recognize processes that are affected by energy quantization and explain how quantization is predicted by the Schrodinger equation