*assignment*Spin Fermi Estimate*assignment*Homework##### Spin Fermi Estimate

Quantum Fundamentals 2023 (2 years) 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.

*assignment*Energy of a relativistic Fermi gas*assignment*Homework##### Energy of a relativistic Fermi gas

Fermi gas Relativity Thermal and Statistical Physics 2020For 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}

*face*Fermi and Bose gases*face*Lecture120 min.

##### Fermi and Bose gases

Thermal and Statistical Physics 2020Fermi level fermion boson Bose gas Bose-Einstein condensate ideal gas statistical mechanics phase transition

These lecture notes from week 7 of Thermal and Statistical Physics apply the grand canonical ensemble to fermion and bosons ideal gasses. They include a few small group activities.*accessibility_new*Spin 1/2 with Arms*accessibility_new*Kinesthetic10 min.

##### Spin 1/2 with Arms

Quantum Fundamentals 2023 (2 years)Quantum State Vectors Complex Numbers Spin 1/2 Arms Representation

Students, working in pairs, use their left arms to represent each component in a two-state quantum spin 1/2 system. Reinforces the idea that quantum states are complex valued vectors. Students make connections between Dirac, matrix, and Arms representation.*assignment*Mass-radius relationship for white dwarfs*assignment*Homework##### Mass-radius relationship for white dwarfs

White dwarf Mass Density Energy Thermal and Statistical Physics 2020Consider 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.

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 \(-\frac53\frac{GM^2}{R}\)).

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.

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

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

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 \(\text{km}\).

*face*Ideal Gas*face*Lecture120 min.

##### Ideal Gas

Thermal and Statistical Physics 2020ideal gas particle in a box grand canonical ensemble chemical potential statistical mechanics

These notes from week 6 of Thermal and Statistical Physics 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.*assignment*Potential vs. Potential Energy*assignment*Homework##### Potential vs. Potential Energy

Static Fields 2023 (6 years)In this course, two of the primary examples we will be using are the potential due to gravity and the potential due to an electric charge. Both of these forces vary like \(\frac{1}{r}\), so they will have many, many similarities. Most of the calculations we do for the one case will be true for the other. But there are some extremely important differences:

- Find the value of the electrostatic potential energy of a system consisting of a hydrogen nucleus and an electron separated by the Bohr radius. Find the value of the gravitational potential energy of the same two particles at the same radius. Use the same system of units in both cases. Compare and the contrast the two answers.
- Find the value of the electrostatic potential due to the nucleus of a hydrogen atom at the Bohr radius. Find the gravitational potential due to the nucleus at the same radius. Use the same system of units in both cases. Compare and contrast the two answers.
- Briefly discuss at least one other fundamental difference between electromagnetic and gravitational systems. Hint: Why are we bound to the earth gravitationally, but not electromagnetically?

*group*Sequential Stern-Gerlach Experiments*group*Small Group Activity10 min.

##### Sequential Stern-Gerlach Experiments

Quantum Fundamentals 2023 (3 years)*accessibility_new*Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems*accessibility_new*Kinesthetic10 min.

##### Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems

Quantum Fundamentals 2023 (2 years)quantum states complex numbers arms Bloch sphere relative phase overall phase

Students, working in pairs, use the Arms representations to represent states of spin 1/2 system. Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT distinguish quantum states) and relative phase (which does distinguish quantum states).*group*Spin-1 Time Evolution*group*Small Group Activity120 min.

##### Spin-1 Time Evolution

Quantum Fundamentals 2023 Students do calculations for time evolution for spin-1.-
Quantum Fundamentals 2023
Two electrons are placed in a magnetic field in the \(z\)-direction. The initial state of the first electron is \(\frac{1}{\sqrt{2}}\begin{pmatrix}
1\\ i\\
\end{pmatrix}\) and the initial state of the second electron is \(\frac{1}{2}\begin{pmatrix}
\sqrt{3}\\ 1\\
\end{pmatrix}\).
- Find the probabilty of measuring each particle to have spin-up in the \(x\)-, \(y\)-, and \(z\)-directions at \(t = 0\).
- Find the probabilty of measuring each particle to have spin-up in the \(x\)-, \(y\)-, and \(z\)-directions at some later time \(t\).
- Calculate the expectation values for \(S_x\), \(S_y\), and \(S_z\) for each particle as functions of time.
- Are there any times when all the probabilities you have calculated are the same as they were at \(t = 0\)?