Consider a system of fixed volume in thermal contact with a resevoir. Show that the mean square fluctuations in the energy of the system is \begin{equation} \left<\left(\varepsilon-\langle\varepsilon\rangle\right)^2\right> = k_BT^2\left(\frac{\partial U}{\partial T}\right)_{V} \end{equation} Here \(U\) is the conventional symbol for \(\langle\varepsilon\rangle\). Hint: Use the partition function \(Z\) to relate \(\left(\frac{\partial U}{\partial T}\right)_V\) to the mean square fluctuation. Also, multiply out the term \((\cdots)^2\).

(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 \(\left<N\right>=0\).

In this unit, you will explore the classical mechanics of central forces, especially gravitational orbits like the earth going around the sun.

Motivating Questions

• What shapes can the orbits have?
• What are Kepler's laws and why are they true?
• What is an effective potential diagram and how can it be used to predict the shape of an orbit?

Key Activities/Problems

• Problem: Undo Formulas for Center of Mass (Geometry)
• Activity: Acting Out Effective Potentials (In class, only)
• Activity: Effective Potentials
• Problem: Hockey
• Problem: Scattering

Unit Learning Outcomes

At the end of this unit, you should be able to:

• List the properties that define a central force system.
• Calculate a reduced mass for a two-body system and describe why it is important.
• Use the solution (algebraic or geometric) to a reduced mass system to describe the motion of the original system.
• Describe the role that conservation of energy and angular momentum play in a central force system. In particular, where do these properties appear in the solutions of the equations of motion?
• Use an effective potential diagram to predict the possible orbits in a central force system: which orbits are bound or unbound? which are closed or open? where will the turning points be?

Equation Sheet for This Unit

• Classical Orbits Equation Sheet

These lecture notes for the second week of https://paradigms.oregonstate.edu/courses/ph441 involve relating entropy and temperature in the microcanonical ensemble, using a paramagnet as an example. These notes include a few small group activities.

These lecture notes for the first week of https://paradigms.oregonstate.edu/courses/ph441 include a couple of small group activities in which students work with the Gibbs formulation of the entropy.

Students examine a plastic “surface” graph of the gravitational potential energy of an Earth-satellite system to explore the properties of gravitational potential energy for a spherically symmetric system.

Students examine a plastic "surface" graph of the gravitational potential energy of a Earth-satellite system to make connections between gravitational force and gravitational potential energy.

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.

Students observe the motion of a puck tethered to the center of the airtable. Then they plot the potential energy for the puck on their small whiteboards. A class discussion follows based on what students have written on their whiteboards.

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.

• How to translate a complicated wavefunction into eigenstates.
• Refresher on how to find expectation values and probabilities in a region.
• How to use the symmetry of the wavefunction to tell you something about measurements.

In quantum mechanics, it turns out that the overall phase for a state does not have any physical significance. Therefore, you will need to become quick at rearranging the phase of various states. For each of the vectors listed below, rewrite the vector as an overall complex phase times a new vector whose first component is real and positive. \[\left|D\right\rangle\doteq \begin{pmatrix} 7e^{i\frac{\pi}{6}}\\ 3e^{i\frac{\pi}{2}}\\ -1\\ \end{pmatrix}\\ \left|E\right\rangle\doteq \begin{pmatrix} i\\ 4\\ \end{pmatrix}\\ \left|F\right\rangle\doteq \begin{pmatrix} 2+2i\\ 3-4i\\ \end{pmatrix} \]

1-D Particle-in-a-box

Hamiltonian: \begin{align} \hat{H} &\doteq -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} \end{align} Eigenstates: \begin{align} \left|{n}\right\rangle &\doteq\sqrt{\frac{2}{L}}\, \sin\frac{n\pi x}{L}\\ n&=\left\{1, 2, 3, \dots\right\} \end{align}

Eigenvalue Equations: \begin{align} \hat{H}\left|{n}\right\rangle &=\frac{\pi^2\hbar^2}{2\mu L^2}\, n^2 \left|{n}\right\rangle \\ \end{align}

Particle-on-a-Ring

Hamiltonian: \begin{align} \hat{H} &\doteq -\frac{\hbar^2}{2I}\frac{\partial^2}{\partial \phi^2} \end{align} Eigenstates: \begin{align} \left|{m}\right\rangle &\doteq\frac{1}{\sqrt{2\pi r_0}}\, e^{im\phi}\\ m&=\left\{\dots -2, -1, 0, 1, 2, \dots\right\} \end{align} Eigenvalue Equations: \begin{align} \hat{H}\left|{m}\right\rangle &=\frac{\hbar^2}{2I}\, m^2 \left|{m}\right\rangle \\ \hat{L}^2\left|{m}\right\rangle &=\hbar^2\, m^2 \left|{m}\right\rangle \\ \hat{L}_z\left|{m}\right\rangle &=\hbar\, m \left|{m}\right\rangle \end{align}

2-D Particle-in-a-Box

Hamiltonian: \begin{align} \hat{H} &\doteq -\frac{\hbar^2}{2m}\Big(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}) \end{align} Eigenstates: \begin{align} \left|{mn}\right\rangle &\doteq\sqrt{\frac{2}{L_x}}\sqrt{\frac{2}{L_y}}\, \sin\frac{m\pi x}{L_x}\sin\frac{n\pi y}{L_y}\\ m&=\left\{1, 2, 3, \dots\right\}\\ n&=\left\{1, 2, 3, \dots\right\} \end{align} Eigenvalue Equations: \begin{align} \hat{H}\left|{mn}\right\rangle &=\frac{\pi^2\hbar^2}{2\mu}\, \left(\frac{m^2}{L_x^2}+\frac{n^2}{L_y^2}\right) \left|{mn}\right\rangle \\ \end{align}

Particle-on-a-Sphere

Hamiltonian: \begin{align} \hat{H} &\doteq -\frac{\hbar^2}{2I} \Big[\frac{1}{\sin\theta}\frac{\partial}{\partial \theta} \Big(\sin\theta \frac{\partial}{\partial \theta} \big) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial \phi^2} \Big] \end{align} Eigenstates: \begin{align} \left|{\ell m}\right\rangle &\doteq Y_{\ell}^m(\theta, \phi)\\ &=(-1)^{\frac{m+|m|}{2}}\sqrt{\frac{2\ell+1}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}} \,P_{\ell}^m(\cos\theta)\, e^{im\phi}\\ \ell&=\left\{0, 1, 2, \dots\right\}\\ m&=\left\{-\ell, \dots , 0, \dots,\ell\right\} \end{align} Eigenvalue Equations: \begin{align} \hat{H}\left|{\ell m}\right\rangle &=\frac{\hbar^2}{2I}\, \ell(\ell+1) \left|{\ell m}\right\rangle \\ \hat{L}^2\left|{\ell m}\right\rangle &=\hbar^2\, \ell(\ell+1) \left|{\ell m}\right\rangle \\ \hat{L}_z\left|{\ell m}\right\rangle &=\hbar\, m \left|{\ell m}\right\rangle \end{align}

Hydrogen Atom

Hamiltonian: \begin{align} \hat{H} &\doteq -\frac{\hbar^2}{2\mu} \nabla^2 - \frac{ke^2}{r} \\ &\doteq -\frac{\hbar^2}{2\mu r^2} \Big[\frac{\partial}{\partial r} \Big( r^2 \frac{\partial}{\partial r} \Big) + \frac{1}{\sin\theta}\frac{\partial}{\partial \theta} \Big(\sin\theta \frac{\partial}{\partial \theta} \big) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial \phi^2} \Big] - \frac{ke^2}{r} \end{align} Eigenstates: \begin{align} \left|{n\ell m}\right\rangle &\doteq R_{n\ell}(r)\, Y_{\ell}^m(\theta, \phi)\\ &=-\sqrt{\left(\frac{2Z}{na_0}\right)^3 \frac{(n-\ell-1)!}{2n[(n+\ell)!]^3}} \left(\frac{2\rho}{n}\right)^{\ell}\, e^{-\frac{\rho}{n}}\, L_{n+\ell}^{2\ell+1}{\scriptstyle{\left(\frac{2\rho}{n}\right)}} (-1)^{\frac{m+|m|}{2}} \sqrt{\frac{2\ell+1}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}} \,P_{\ell}^m(\cos\theta)\, e^{im\phi}\\ \rho&=\frac{Zr}{a_0}\\ n&=\left\{1, 2, 3,\dots\right\}\\ \ell&=\left\{0, 1, 2, \dots, n-1\right\}\\ m&=\left\{-\ell, \dots , 0, \dots,\ell\right\} \end{align} Eigenvalue Equations: \begin{align} \hat{H}\left|{n\ell m}\right\rangle &=-\frac{1}{2}\left(\frac{Ze^2}{4\pi\epsilon_0}\right)^2 \frac{\mu}{\hbar^2}\,\frac{1}{n^2}\, \left|{n \ell m}\right\rangle \\ &=-13.6 \text{eV}\,\frac{1}{n^2}\, \left|{n \ell m}\right\rangle \\ \hat{L}^2\left|{n \ell m}\right\rangle &=\hbar^2\, \ell(\ell+1) \left|{n \ell m}\right\rangle \\ \hat{L}_z\left|{n \ell m}\right\rangle &=\hbar\, m \left|{n \ell m}\right\rangle \end{align}

Show that if a linear combination of ring energy eigenstates is normalized, then the coefficients must satisfy \begin{equation} \sum_{m=-\infty}^{\infty} \vert c_m\vert^2=1 \end{equation}

You know that the normalized spatial eigenfunctions for a particle in a 1-D box of length \(L\) are \(\sqrt{\frac{2}{L}}\sin{\frac{n\pi x}{L}}\). If you want the eigenfunctions for a particle in a 2-D box, then you just multiply together the eigenfunctions for a 1-D box in each direction. (This is what the separation of variables procedure tells you to do.)

1. Find the normalized eigenfunctions for a particle in a 2-D box with sides of length \(L\) in the \(x\)-direction and length \(W\) in the \(y\)-direction.
2. Find the Hamiltonian for a 2-D box and show that your eigenstates are indeed eigenstates and find a formula for the possible energies

3. Any sufficiently smooth spatial wave function inside a 2-D box can be expanded in a double sum of the product wave functions, i.e. \begin{equation} \psi(x,y)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\; \hbox{eigenfunction}_n(x)\;\hbox{eigenfunction}_m(y) \end{equation} Using your expressions from part (a) above, write out all the terms in this sum out to \(n=3\), \(m=3\). Arrange the terms, conventionally, in terms of increasing energy.

   You may find it easier to work in bra/ket notation: \begin{align*} \left|{\psi}\right\rangle &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{n}\right\rangle \left|{m}\right\rangle \\ &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{nm}\right\rangle \end{align*}

4. Find a formula for the \(c_{nm}\)s in part (c). Find the formula first in bra ket notation and then rewrite it in wave function notation.

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.”

Expectation Values and Uncertainty

You have a system that consists of quantum particles with spin. On this system, you will perform a Stern-Gerlach experiment with an analyzer oriented in the \(z\)-direction.

Consider one of the different initial spin states described below:

A spin 1/2 particle described by:

1. \(\left|{+}\right\rangle \)
2. \(\frac{i}{2}\left|{+}\right\rangle -\frac{\sqrt{3}}{2}\left|{-}\right\rangle \)

3. \(\left|{+}\right\rangle _x\)

   A spin 1 particle described by:

4. \(\left|{0}\right\rangle \)
5. \(\left|{-1}\right\rangle _x\)
6. \(\frac{2}{3}\left|{1}\right\rangle +\frac{i}{3}\left|{0}\right\rangle -\frac{2}{3}\left|{-1}\right\rangle \)

• List the possible values of spin you could measure and determine the probability associated with each value of the z-component of spin.

  
  

• Plot a histogram of the probabilities.

  
  

• Find the expectation value of the z-component of spin.

  
  

• Find the uncertainty of the z-component of spin.

Introduction

I like to break this activity into two parts:

(1) Calculating expectation values and relating them to the associated distributions of the probabilities of results, and

(2) Calculating the quantum uncertainty of the state and relating the uncertainty to distributions of the probabilities of results.

Therefore, I have my students do the first part of the activity before I introduce quantum uncertainty.

I introduce the activity by reminding students about two ways of calculating the expectation value. Given a quantum state \(\left|{\psi}\right\rangle \), for a measurement of an observable represented by an operator \(\hat{A}\) with eigenstates \(\left|{a_i}\right\rangle \) and eigenvalues \(a_i\): \begin{align*} \langle \hat{A} \rangle &= \sum_{i} a_i\mathcal{P}(a_i) \\ &= \left\langle {\psi}\right|\hat{A}\left|{\psi}\right\rangle \end{align*}

After the students calculate expectation values and we have a whole class discussion about 1 of the examples, then I do a lecture introducing the quantity of quantum uncertainty (relating it to the standard deviation of the distribution of probabilities by spin component value) and deriving the simplified equation:

\[\Delta A = \sqrt{\langle A^2 \rangle - \langle A \rangle ^2}\]

Student Conversations

1. One could have each group report out, or the instructor could discuss a few key examples.

   For expectation value, I like to talk about Case 2: \(\frac{i}{2}\left|{+}\right\rangle -\frac{\sqrt{3}}{2}\left|{-}\right\rangle \), where the probabilities of the two outcomes are not equal to show how the weighting plays out. Also, the expectation value is not a possible measurement value, and I like to talk about that. “Expectation” value is a misleading name for this quantity - it characterizes the distribution and is not necessarily a result of an individual measurement.

   I also like to discuss an example like Case 5: \(\left|{1}\right\rangle _x\) where the distribution is symmetric around \(0\hbar\).

2. I think it's important to encourage students to calculate expectation values both ways (with probabilities and as a bracket with matrix notation) while the teaching team is available to help them.

3. For quantum uncertainty, I like to talk about an example like Case 3: \(\left|{+}\right\rangle _x\) where all the individual measurements are the same ”distance” away from the expectation value as a sensemaking exercise to connect to a conceptual interpretation of physics.

   I also like to discuss an example like Case 5: \(\left|{-1}\right\rangle _x\), where the fact that we're taking an rms average is apparent: half the measurements are \(\hbar\) away from the expectation value and the other half are \(0\hbar\) away, but the uncertainty is \(\hbar/\sqrt{2}\).

Students are asked to find eigenvalues, probabilities, and expectation values for \(H\), \(L^2\), and \(L_z\) for a superposition of \(\vert n \ell m \rangle\) states. This can be done on small whiteboards or with the students working in groups on large whiteboards.

Students then work together in small groups to find the matrices that correspond to \(H\), \(L^2\), and \(L_z\) and to redo \(\langle E\rangle\) in matrix notation.

Students use Mathematica to visualize the probability density distribution for the hydrogen atom orbitals with the option to vary the values of \(n\), \(\ell\), and \(m\).

Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.

Students work in small groups to use completeness relations to change the basis of quantum states.

• How to form a state as a column vector in matrix representation.
• How to do probability calculations on all three representations used for quantum systems in PH426.
• How to find probabilities for and the resultant state after measuring degenerate eigenvalues.

In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.