For electrons with an energy \(\varepsilon\gg mc^2\), where
\(m\) is the mass of the electron, the energy is given by
\(\varepsilon\approx pc\) where \(p\) is the momentum. For electrons
in a cube of volume \(V=L^3\) the momentum takes the same values as
for a non-relativistic particle in a box.
Show that in this extreme relativistic limit the Fermi energy of a
gas of \(N\) electrons is given by \begin{align}
\varepsilon_F &= \hbar\pi c\left(\frac{3n}{\pi}\right)^{\frac13}
\end{align} where \(n\equiv \frac{N}{V}\) is the number density.
Show that the total energy of the ground state of the gas is
\begin{align}
U_0 &= \frac34 N\varepsilon_F
\end{align}
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
\(T\rightarrow 0\).
(K&K 7.11) Show for a single
orbital of a fermion system that \begin{align}
\left<(\Delta N)^2\right> = \left<N\right>(1+\left<N\right>)
\end{align} if \(\left<N\right>\) is the average number of fermions in
that orbital. Notice that the fluctuation vanishes for orbitals with
energies far enough from the chemical potential \(\mu\) so that
\(\left<N\right>=1\) or \(\left<N\right>=0\).
These lecture notes from week 7 of https://paradigms.oregonstate.edu/courses/ph441 apply the grand canonical ensemble to fermion and bosons ideal gasses. They include a few small group activities.
Derivative of Fermi-Dirac function Show that the magnitude of the slope of the Fermi-Direc function \(f\) evaluated at the Fermi
level \(\varepsilon =\mu\) is inversely proportional to its temperature. This means that at lower temperatures the Fermi-Dirac
function becomes dramatically steeper.
The following two problems ask you to make Fermi estimates. In a good Fermi estimate, you start from basic scientific facts you already know or quantities that you can reasonably estimate based on your life experiences and then reason your way to estimate a quantity that you would not be able guess. You may look up useful conversion factors or constants. Use words, pictures, and equations to explain your reasoning:
Imagine that you send a pea-sized bead of silver through a Stern-Gerlach device oriented to measure the z-component of intrinsic spin. Estimate the total z-component of the intrinsic spin of the ball you would measure in the HIGHLY improbable case that every atom is spin up.
Protons, neutrons, and electrons are all spin-1/2 particles. Give a (very crude) order of magnitude estimate of the number of these particles in your body.
Consider one mole of an
ideal monatomic gas at 300K and 1 atm. First, let the gas expand
isothermally and reversibly to twice the initial volume; second, let
this be followed by an isentropic expansion from twice to four times
the original volume.
How much heat (in joules) is added to the gas in each of these two
processes?
What is the temperature at the end of the second process?
Suppose the first process is replaced by an irreversible expansion
into a vacuum, to a total volume twice the initial volume. What is
the increase of entropy in the irreversible expansion, in J/K?
Consider an ideal gas of
\(N\) particles, each of mass \(M\), confined to a one-dimensional
line of length \(L\). The particles have spin zero (so you can ignore
spin) and do not interact with one another. Find the entropy at
temperature \(T\). You may assume that the temperature is high enough
that \(k_B T\) is much greater than the ground state energy of one
particle.
These notes from week 6 of https://paradigms.oregonstate.edu/courses/ph441 cover the ideal gas from a grand canonical standpoint starting with the solutions to a particle in a three-dimensional box. They include a number of small group activities.
These notes, from the third week of https://paradigms.oregonstate.edu/courses/ph441 cover the canonical ensemble and Helmholtz free energy. They include a number of small group activities.
Consider one particle
confined to a cube of side \(L\); the concentration in effect is
\(n=L^{-3}\). Find the kinetic energy of the particle when in the
ground state. There will be a value of the concentration for which
this zero-point quantum kinetic energy is equal to the temperature
\(kT\). (At this concentration the occupancy of the lowest orbital is
of the order of unity; the lowest orbital always has a higher
occupancy than any other orbital.) Show that the concentration \(n_0\)
thus defined is equal to the quantum concentration \(n_Q\) defined by
(63): \begin{equation}
n_Q \equiv \left(\frac{MkT}{2\pi\hbar^2}\right)^{\frac32}
\end{equation} within a factor of the order of unity.
Consider a column of atoms each of mass \(M\) at temperature \(T\) in
a uniform gravitational field \(g\). Find the thermal average
potential energy per atom. The thermal average kinetic energy is
independent of height. Find the total heat capacity per atom. The
total heat capacity is the sum of contributions from the kinetic
energy and from the potential energy. Take the zero of the
gravitational energy at the bottom \(h=0\) of the column. Integrate
from \(h=0\) to \(h=\infty\). You may assume the gas is ideal.
Students struggle with understanding that entropy can be created. It's an extensive quantity, and is the only one that isn't normally conserved, so that makes it pretty weird. We (professors) don't always realize how very weird this is, and students don't have the vocabulary to explain it to us, and are often afraid to try.