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 Lecture
120 min.
Planck distribution blackbody radiation photon statistical mechanics
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 Small Group Activity
30 min.
group Small Group Activity
30 min.
group Small Group Activity
30 min.
group Small Group Activity
60 min.
group Small Group Activity
30 min.
assignment Homework
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 \(\frac1{V^{\frac23}}\).
Find an expression for the entropy of a Fermi electron gas in the region \(kT\ll \varepsilon_F\). Notice that \(S\rightarrow 0\) as \(T\rightarrow 0\).
group Small Group Activity
30 min.
assignment Homework
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).
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 \(T_C\)).
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?
Does the work done on the two isentropic stages cancel each other, as for the ideal gas?
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.