## Activity: Energy radiated from one oscillator

Contemporary Challenges 2022 (3 years)
This lecture is one step in motivating the form of the Planck distribution.
• Media
follows

## Classical prediction

\begin{align} \left<\text{energy rate}\right> &= \frac23 \frac{q^2}{4\pi\epsilon_0 c^3}\frac{k_BT}{m}\omega_0^2 \end{align} Draw a graph with $\left<\text{energy rate}\right>$ (the optical power output) on the vertical axis, and the oscillator frequency (or alternatively $\hbar\omega_0$) on the horizontal axis. Sketch the relationship when
1. $T=300\text{ K}$
2. $T=600\text{ K}$
Assume the mass and charge of the the oscillators are constants.

## Quantum prediction

When we account for energy quanta, the expression changes to \begin{align} \left<\text{energy rate}\right> &= \begin{cases} \frac23 \frac{q^2}{4\pi\epsilon_0 c^3}\frac{k_BT}{m}\omega_0^2 & \text{when }\hbar\omega\ll k_BT \\ \frac{\hbar\omega^3}{4\pi^2c^2}e^{-\frac{\hbar\omega}{k_BT}} & \text{when }\hbar\omega\gg k_BT \end{cases} \end{align} Sketch the relationship on your same graph, when
1. $T=300\text{ K}$
2. $T=600\text{ K}$
Assume the mass and charge of the the oscillators are constants.
• face Wavelength of peak intensity

face Lecture

5 min.

##### Wavelength of peak intensity
Contemporary Challenges 2022 (2 years)

This very short lecture introduces Wein's displacement law.
• face Equipartition theorem

face Lecture

30 min.

##### Equipartition theorem
Contemporary Challenges 2022 (3 years)

This lecture introduces the equipartition theorem.
• group Optical depth of atmosphere

group Small Group Activity

30 min.

##### Optical depth of atmosphere
Contemporary Challenges 2022 (3 years) In this activity students estimate the optical depth of the atmosphere at the infrared wavelength where carbon dioxide has peak absorption.
• face Basics of heat engines

face Lecture

10 min.

##### Basics of heat engines
Contemporary Challenges 2022 (3 years) This brief lecture covers the basics of heat engines.
• group Earthquake waves

group Small Group Activity

30 min.

##### Earthquake waves
Contemporary Challenges 2022 (3 years)

In this activity students use the known speed of earthquake waves to estimate the Young's modulus of the Earth's crust.
• group Thermal radiation at twice the temperature

group Small Group Activity

10 min.

##### Thermal radiation at twice the temperature
Contemporary Challenges 2022 (3 years)

This small group activity has students reasoning about how the Planck distribution shifts when the temperature is doubled. This leads to a qualitative argument for the Stefan-Boltzmann law.
• 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.

• group Black space capsule

group Small Group Activity

30 min.

##### Black space capsule
Contemporary Challenges 2022 (2 years)

In this activity, students apply the Stefan-Boltzmann equation and the principle of energy balance in steady state to find the steady state temperature of a black object in near-Earth orbit.
• group Hydrogen emission

group Small Group Activity

30 min.

##### Hydrogen emission
Contemporary Challenges 2022 (4 years)

In this activity students work out energy level transitions in hydrogen that lead to visible light.
• group de Broglie wavelength after freefall

group Small Group Activity

30 min.

##### de Broglie wavelength after freefall
Contemporary Challenges 2022 (3 years)

In this activity students combine energy conservation with the relationship between the de Broglie wavelength and momentum to find the wavelength of atoms that have been dropped a given distance.

Learning Outcomes