## Zapping With d 1

• 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.

• assignment Quantum harmonic oscillator

assignment Homework

##### Quantum harmonic oscillator
Entropy Quantum harmonic oscillator Frequency Energy Thermal and Statistical Physics 2020
1. Find the entropy of a set of $N$ oscillators of frequency $\omega$ as a function of the total quantum number $n$. Use the multiplicity function: $$g(N,n) = \frac{(N+n-1)!}{n!(N-1)!}$$ and assume that $N\gg 1$. This means you can make the Sitrling approximation that $\log N! \approx N\log N - N$. It also means that $N-1 \approx N$.

2. Let $U$ denote the total energy $n\hbar\omega$ of the oscillators. Express the entropy as $S(U,N)$. Show that the total energy at temperature $T$ is $$U = \frac{N\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1}$$ This is the Planck result found the hard way. We will get to the easy way soon, and you will never again need to work with a multiplicity function like this.

• assignment Free energy of a harmonic oscillator

assignment Homework

##### Free energy of a harmonic oscillator
Helmholtz free energy harmonic oscillator Thermal and Statistical Physics 2020

A one-dimensional harmonic oscillator has an infinite series of equally spaced energy states, with $\varepsilon_n = n\hbar\omega$, where $n$ is an integer $\ge 0$, and $\omega$ is the classical frequency of the oscillator. We have chosen the zero of energy at the state $n=0$ which we can get away with here, but is not actually the zero of energy! To find the true energy we would have to add a $\frac12\hbar\omega$ for each oscillator.

1. Show that for a harmonic oscillator the free energy is $$F = k_BT\log\left(1 - e^{-\frac{\hbar\omega}{k_BT}}\right)$$ Note that at high temperatures such that $k_BT\gg\hbar\omega$ we may expand the argument of the logarithm to obtain $F\approx k_BT\log\left(\frac{\hbar\omega}{kT}\right)$.

2. From the free energy above, show that the entropy is $$\frac{S}{k_B} = \frac{\frac{\hbar\omega}{kT}}{e^{\frac{\hbar\omega}{kT}}-1} - \log\left(1-e^{-\frac{\hbar\omega}{kT}}\right)$$

This entropy is shown in the nearby figure, as well as the heat capacity.

• 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.
• assignment Centrifuge

assignment Homework

##### Centrifuge
Centrifugal potential Thermal and Statistical Physics 2020 A circular cylinder of radius $R$ rotates about the long axis with angular velocity $\omega$. The cylinder contains an ideal gas of atoms of mass $M$ at temperature $T$. Find an expression for the dependence of the concentration $n(r)$ on the radial distance $r$ from the axis, in terms of $n(0)$ on the axis. Take $\mu$ as for an ideal gas.
• group Energy radiated from one oscillator

group Small Group Activity

30 min.

##### Energy radiated from one oscillator
Contemporary Challenges 2022 (4 years)

This lecture is one step in motivating the form of the Planck distribution.
• face Guide to Professional Typography in Physics

face Lecture

5 min.

##### Guide to Professional Typography in Physics
Contemporary Challenges 2022 (4 years)

This is really a handout, which gives students guidelines on how to type up physics content.
• group Fourier Transform of a Plane Wave

group Small Group Activity

5 min.

##### Fourier Transform of a Plane Wave
Periodic Systems 2022

Fourier Transforms and Wave Packets

• accessibility_new Using Arms to Represent Time Dependence in Spin 1/2 Systems

accessibility_new Kinesthetic

10 min.

##### Using Arms to Represent Time Dependence in Spin 1/2 Systems
Quantum Fundamentals 2022 (2 years)

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.
• face Quantum Reference Sheet

face Lecture

5 min.

##### Quantum Reference Sheet
Central Forces 2023 (6 years)
• Energy and Entropy 2021 (2 years)

Find the differential of each of the following expressions; zap each of the following with $d$:

1. $f=3x-5z^2+2xy$

2. $g=\frac{c^{1/2}b}{a^2}$

3. $h=\sin^2(\omega t)$

4. $j=a^x$

5. $k=5 \tan\left(\ln{\left(\frac{V_1}{V_2}\right)}\right)$