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Activities

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.

  1. 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.

  2. Show that the total energy of the ground state of the gas is \begin{align} U_0 &= \frac34 N\varepsilon_F \end{align}

Consider a white dwarf of mass \(M\) and radius \(R\). The dwarf consists of ionized hydrogen, thus a bunch of free electrons and protons, each of which are fermions. Let the electrons be degenerate but nonrelativistic; the protons are nondegenerate.

  1. Show that the order of magnitude of the gravitational self-energy is \(-\frac{GM^2}{R}\), where \(G\) is the gravitational constant. (If the mass density is constant within the sphere of radius \(R\), the exact potential energy is \(-\frac53\frac{GM^2}{R}\)).

  2. Show that the order of magnitude of the kinetic energy of the electrons in the ground state is \begin{align} \frac{\hbar^2N^{\frac53}}{mR^2} \approx \frac{\hbar^2M^{\frac53}}{mM_H^{\frac53}R^2} \end{align} where \(m\) is the mass of an electron and \(M_H\) is the mas of a proton.

  3. Show that if the gravitational and kinetic energies are of the same order of magnitude (as required by the virial theorem of mechanics), \(M^{\frac13}R \approx 10^{20} \text{g}^{\frac13}\text{cm}\).

  4. If the mass is equal to that of the Sun (\(2\times 10^{33}g\)), what is the density of the white dwarf?

  5. It is believed that pulsars are stars composed of a cold degenerate gas of neutrons (i.e. neutron stars). Show that for a neutron star \(M^{\frac13}R \approx 10^{17}\text{g}^{\frac13}\text{cm}\). What is the value of the radius for a neutron star with a mass equal to that of the Sun? Express the result in \(\text{km}\).

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.

In this entire problem, keep results to first order in the van der Waals correction terms \(a\) and $b.

  1. Show that the entropy of the van der Waals gas is \begin{align} S &= Nk\left\{\ln\left(\frac{n_Q(V-Nb)}{N}\right)+\frac52\right\} \end{align}

  2. Show that the energy is \begin{align} U &= \frac32 NkT - \frac{N^2a}{V} \end{align}

  3. Show that the enthalpy \(H\equiv U+pV\) is \begin{align} H(T,V) &= \frac52NkT + \frac{N^2bkT}{V} - 2\frac{N^2a}{V} \\ H(T,p) &= \frac52NkT + Nbp - \frac{2Nap}{kT} \end{align}

Effects of High Altitude by Randall Munroe, at xkcd.

Small Group Activity

30 min.

Hydrogen emission
In this activity students work out energy level transitions in hydrogen that lead to visible light.

Computational Activity

120 min.

Kinetic energy
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.

Small Group Activity

30 min.

Using \(pV\) and \(TS\) Plots
Students work out heat and work for rectangular paths on \(pV\) and \(TS\) plots. This gives with computing heat and work, applying the First Law, and recognizing that internal energy is a state function, which cannot change after a cyclic process.

Lecture

5 min.

Energy and Entropy review
This very quick lecture reviews the content taught in https://paradigms.oregonstate.edu/courses/ph423, and is the first content in https://paradigms.oregonstate.edu/courses/ph441.

Small Group Activity

30 min.

A glass of water
Students generate a list of properties a glass of water might have. The class then discusses and categorizes those properties.

Small Group Activity

30 min.

Mass is not Conserved

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?

Small Group Activity

30 min.

Heat capacity of N2
Students sketch the temperature-dependent heat capacity of molecular nitrogen. They apply the equipartition theorem and compute the temperatures at which degrees of freedom “freeze out.”
The instructor gives a brief lecture about time dependence of energy eigenstates (e.g. McIntyre, 3.1). Notes for the students are attached.

Small Group Activity

120 min.

Equipotential Surfaces
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.

Small Group Activity

30 min.

Work By An Electric Field (Contour Map)
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.

Small Group Activity

30 min.

Expectation Values for a Particle on a Ring
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 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.

Small Group Activity

30 min.

Electric Field of Two Charged Plates
  • 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
    1. objects with like charge repel and opposite charge attract,
    2. object tend to move toward lower energy configurations
    3. The potential energy of a charged particle is related to its charge: \(U=qV\)
    4. The force on a charged particle is related to its charge: \(\vec{F}=q\vec{E}\)

Small Group Activity

30 min.

Superposition States for a Particle on a Ring
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.

Small Group Activity

60 min.

Ice Calorimetry Lab
The students will set up a Styrofoam cup with heating element and a thermometer in it. They will measure the temperature as a function of time, and thus the energy transferred from the power supply, from which they compute changes in entropy.

Small Group Activity

30 min.

Covariation in Thermal Systems
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.

Small Group Activity

30 min.

Heat and Temperature of Water Vapor
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.

Small Group Activity

30 min.

Black space capsule
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.

Small Group Activity

30 min.

Applying the equipartition theorem
Students count the quadratic degrees of freedom of a few toy molecules to predict their internal energy at temperature \(T\).

Small Group Activity

30 min.

de Broglie wavelength after freefall
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.