Find the equilibrium value at temperature \(T\)
of the fractional magnetization \begin{equation}
\frac{\mu_{tot}}{Nm} \equiv \frac{2\langle s\rangle}{N}
\end{equation} of a system of \(N\) spins each of magnetic moment
\(m\) in a magnetic field \(B\). The spin excess is \(2s\). The energy
of this system is given by \begin{align}
U &= -\mu_{tot}B
\end{align} where \(\mu_{tot}\) is the total magnetization. Take the
entropy as the logarithm of the multiplicity \(g(N,s)\) as given in
(1.35 in the text): \begin{equation}
S(s) \approx k_B\log g(N,0) - k_B\frac{2s^2}{N}
\end{equation} for \(|s|\ll N\), where \(s\) is the spin excess, which
is related to the magnetization by \(\mu_{tot} = 2sm\). Hint:
Show that in this approximation \begin{equation}
S(U) = S_0 - k_B\frac{U^2}{2m^2B^2N},
\end{equation} with \(S_0=k_B\log g(N,0)\). Further, show that
\(\frac1{kT} = -\frac{U}{m^2B^2N}\), where \(U\) denotes
\(\langle U\rangle\), the thermal average energy.
Consider a system that may be unoccupied with energy zero, or
occupied by one particle in either of two states, one of energy zero
and one of energy \(\varepsilon\). Find the Gibbs sum for this
system is in terms of the activity \(\lambda\equiv e^{\beta\mu}\).
Note that the system can hold a maximum of one particle.
Solve for the thermal average occupancy of the system in terms of
\(\lambda\).
Show that the thermal average occupancy of the state at energy
\(\varepsilon\) is \begin{align}
\langle N(\varepsilon)\rangle =
\frac{\lambda e^{-\frac{\varepsilon}{kT}}}{\mathcal{Z}}
\end{align}
Find an expression for the thermal average energy of the system.
Allow the possibility that the orbitals at \(0\) and at
\(\varepsilon\) may each be occupied each by one particle at the
same time; Show that \begin{align}
\mathcal{Z} &= 1 + \lambda + \lambda e^{-\frac{\varepsilon}{kT}} +
\lambda^2 e^{-\frac{\varepsilon}{kT}}
\\
&= (1+\lambda)\left(1+e^{-\frac{\varepsilon}{kT}}\right)
\end{align} Because \(\mathcal{Z}\) can be factored as shown, we
have in effect two independent systems.
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.
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}
Groups are asked to analyze the following standard problem:
Two identical lumps of clay of (rest) mass m collide head on, with each
moving at 3/5 the speed of light. What is the mass of the resulting lump of
clay?
Students implement a finite-difference approximation for the kinetic energy operator as a matrix, and then use numpy to solve for eigenvalues and eigenstates, which they visualize.
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.
Students will estimate the work done by a given electric field. They will connect the work done to the height of a plastic surface graph of the electric potential.
Students need to understand that the surface represents the electric potential in the center of a parallel plate capacitor. Try doing the activity Electric Potential of Two Charged Plates before this activity.
Students should know that
objects with like charge repel and opposite charge attract,
object tend to move toward lower energy configurations
The potential energy of a charged particle is related to its charge: \(U=qV\)
The force on a charged particle is related to its charge: \(\vec{F}=q\vec{E}\)
Students calculate probabilities for energy, angular momentum, and position as a function of time for an initial state that is a linear combination of energy/angular momentum eigenstates for a particle confined to a ring written in bra-ket notation. This activity helps students build an understanding of when they can expect a quantity to depend on time and to give them more practice moving between representations.
Students calculate the expectation value of energy and angular momentum as a function of time for an initial state for a particle on a ring. This state is a linear combination of energy/angular momentum eigenstates written in bra-ket notation.
Students are prompted to consider the scalar superposition of the electric potential due to multiple point charges. First a single point charge is discussed, then four positive charges, then an electric quadrupole. Students draw the equipotential curves in the plane of the charges, while also considering the 3D nature of equipotentials.
In this introduction to heat capacity, students determine a derivative that indicates how much the internal energy changes as the temperature changes when volume is held constant.
Students calculate probabilities for a particle on a ring whose wavefunction is not easily separated into eigenstates by inspection. To find the energy, angular momentum, and position probabilities, students perform integrations with the wavefunction or decompose the wavefunction into a superposition of eigenfunctions.
This lab gives students a chance to take data on the first day of class (or later, but I prefer to do it the first day of class). It provides an immediate context for thermodynamics, and also gives them a chance to experimentally measure a change in entropy. Students are required to measure the energy required to melt ice and raise the temperature of water, and measure the change in entropy by integrating the heat capacity.
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.
Students consider how changing the volume of a system changes the internal energy of the system. Students use plastic graph models to explore these functions.
