format_list_numbered Sequence

Gradient Sequence
This sequence starts with an introduction to partial derivatives and continues through gradient. While some of the activities/problems are pure math, a number of other activities/problems are situated in the context of electrostatics. This sequence is intended to be used intermittently across multiple days or even weeks of a course or even multiple courses.

face Lecture

30 min.

Introducing entropy
This lecture introduces the idea of entropy, including the relationship between entropy and multiplicity as well as the relationship between changes in entropy and heat.

Scan and upload your answers to Gradescope using the Gradescope tab in Canvas. You can use the scanner in Weniger 304F, or a scanning app on a cell phone, or make a pdf directly from a tablet computer; Gradescope offers advice and suggested apps at this URL. The preferred format is PDF. Photos or JPEG scans are less easy to read (and much larger), and should be used only if no alternative is available.

On Gradescope, you will be prompted to associate submitted pages with problem numbers by selecting pages on the right and questions on the left. You may associate multiple problems with the same page if appropriate.

If you have a problems with this process, please contact your instructor or teaching assistant.

  • Found in: Contemporary Challenges course(s)

format_list_numbered Sequence

Arms Sequence for Complex Numbers and Quantum States

“Arms” is an engaging representation of complex numbers. Students use their left arms to geometrically represent numbers in the complex plane (an Argand diagram).

The sequence starts with pure math activities in which students represent a single complex number (using prompts in both rectangular and exponential forms), demonstrate multiplication of complex numbers in exponential form, and act out a number of different linear transformation on pairs of complex numbers. Later activities, relevant to spin 1/2 systems in quantum mechanics, explore overall phases, relative phases, and time dependence.

These activities can be combined and sequenced in many different ways; see the Instructor's Guides for how to introduce the Arms representation the first time you use it.

First complete the problem Diagonalization. In that notation:
  1. Find the matrix \(S\) whose columns are \(|\alpha\rangle\) and \(|\beta\rangle\). Show that \(S^{\dagger}=S^{-1}\) by calculating \(S^{\dagger}\) and multiplying it by \(S\). (Does the order of multiplication matter?)
  2. Calculate \(B=S^{-1} C S\). How is the matrix \(E\) related to \(B\) and \(C\)? The transformation that you have just done is an example of a “change of basis”, sometimes called a “similarity transformation.” When the result of a change of basis is a diagonal matrix, the process is called diagonalization.
  • Found in: Quantum Fundamentals course(s)

accessibility_new Kinesthetic

10 min.

Using Arms to Visualize Complex Numbers (MathBits)
Students move their left arm in a circle to trace out the complex plane (Argand diagram). They then explore the rectangular and exponential representations of complex numbers by using their left arm to show given complex numbers on the complex plane. Finally they enact multiplication of complex numbers in exponential form and complex conjugation.
  1. Find the entropy of a set of \(N\) oscillators of frequency \(\omega\) as a function of the total quantum number \(n\). Use the multiplicity function: \begin{equation} g(N,n) = \frac{(N+n-1)!}{n!(N-1)!} \end{equation} and assume that \(N\gg 1\). This means you can make the Sitrling approximation that \(\log N! \approx N\log N - N\). It also means that \(N-1 \approx N\).

  2. Let \(U\) denote the total energy \(n\hbar\omega\) of the oscillators. Express the entropy as \(S(U,N)\). Show that the total energy at temperature \(T\) is \begin{equation} U = \frac{N\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1} \end{equation} This is the Planck result found the hard way. We will get to the easy way soon, and you will never again need to work with a multiplicity function like this.

  • Found in: Thermal and Statistical Physics course(s)

group Small Group Activity

10 min.

Gaussian Parameters
Students use an applet to explore the role of the parameters \(N\), \(x_o\), and \(\sigma\) in the shape of a Gaussian \begin{equation} f(x)=Ne^{-\frac{(x-x_0)^2}{2\sigma^2}} \end{equation}
  • Found in: Periodic Systems course(s) Found in: Fourier Transforms and Wave Packets sequence(s)

Suppose \(g(U) = CU^{3N/2}\), where \(C\) is a constant and \(N\) is the number of particles.

  1. Show that \(U=\frac32 N k_BT\).

  2. Show that \(\left(\frac{\partial^2S}{\partial U^2}\right)_N\) is negative. This form of \(g(U)\) actually applies to a monatomic ideal gas.

  • Found in: Thermal and Statistical Physics course(s)

Task: Draw a right triangle. Put a circle around the right angle, that is, the angle that is \(\frac\pi2\) radians.

Preparing your submission:

  • Complete the assignment using your choice of technology. You may write your answers on paper, write them electronically (for instance using xournal), or typeset them (for instance using LaTeX).
  • If using software, please export to PDF. If writing by hand, please scan your work using the AIMS scanner if possible. You can also use a scanning app; Gradescope offers advice and suggested apps at this URL. The preferred format is PDF; photos or JPEG scans are less easy to read (and much larger), and should be used only if no alternative is available.)
  • Please make sure that your file name includes your own name and the number of the assignment, such as "Tevian2.pdf."

Using Gradescope: We will arrange for you to have a Gradescope account, after which you should receive access instructions directly from them. To submit an assignment:

  1. Navigate to https://paradigms.oregonstate.edu and login
  2. Select the appropriate course, such as "AIMS F21". (There will likely be only one course listed.)
  3. Select the assignment called "Sample Assignment"
  4. Follow the instructions to upload your assignment. (The preferred format is PDF.)
  5. You will then be prompted to associate submitted pages with problem numbers by selecting pages on the right and questions on the left. (In this assignment, there is only one of each.) You may associate multiple problems with the same page if appropriate.
  6. When you are finished, click "Submit"
  7. After the assignments have been marked, you can log back in to see instructor comments.

  • Found in: AIMS Maxwell course(s)



Let \begin{align} \boldsymbol{\vec a} &= \boldsymbol{\hat x}-3\boldsymbol{\hat y}-\boldsymbol{\hat z}\\ \boldsymbol{\vec b} &= \boldsymbol{\hat x}+\boldsymbol{\hat y}+2\boldsymbol{\hat z}\\ {\boldsymbol{\vec c}} &= -2\boldsymbol{\hat x}-\boldsymbol{\hat y}+\boldsymbol{\hat z}\\ \boldsymbol{\vec d} &= -\boldsymbol{\hat x}-\boldsymbol{\hat y}+\boldsymbol{\hat z} \end{align}

Which pairs (if any) of these vectors

  1. Are perpendicular?
  2. Are parallel?
  3. Have an angle less than \(\pi/2\) between them?
  4. Have an angle of more than \(\pi/2\) between them?

  • Found in: AIMS Maxwell, Static Fields course(s)

The general solution of the homogeneous differential equation

\[\ddot{x}-\dot{x}-6 x=0\]


\[x(t)=A\, e^{3t}+ B\, e^{-2t}\]

where \(A\) and \(B\) are arbitrary constants that would be determined by the initial conditions of the problem.

  1. Find a particular solution of the inhomogeneous differential equation \(\ddot{x}-\dot{x}-6 x=-25\sin(4 t)\).

  2. Find the general solution of \(\ddot{x}-\dot{x}-6 x=-25\sin(4 t)\).

  3. Some terms in your general solution have an undetermined coefficients, while some coefficients are fully determined. Explain what is different about these two cases.

  4. Find a particular solution of \(\ddot{x}-\dot{x}-6 x=12 e^{-3 t}\)

  5. Find the general solution of \(\ddot{x}-\dot{x}-6 x=12 e^{-3 t}-25\sin(4 t)\)

    How is this general solution related to the particular solutions you found in the previous parts of this question?

    Can you add these particular solutions together with arbitrary coefficients to get a new particular solution?

  6. Sense-making: Check your answer; Explicitly plug in your final answer in part (e) and check that it satisfies the differential equation.

  • Found in: None, Oscillations and Waves course(s)

face Lecture

120 min.

Entropy and Temperature
These lecture notes for the second week of Thermal and Statistical Physics involve relating entropy and temperature in the microcanonical ensemble, using a paramagnet as an example. These notes include a few small group activities.

accessibility_new Kinesthetic

30 min.

Inner Products with Arms
Students perform an inner product between two spin states with the arms representation.

accessibility_new Kinesthetic

30 min.

Inner Product of Spin-1/2 System with Arms
Students use their arms to act out two spin-1/2 quantum states and their inner product.

group Small Group Activity

30 min.

Vector Integrals (Contour Map)
Students explore path integrals using a vector field map and thinking about integration as chop-multiply-add.

group Small Group Activity

30 min.

The Hot Plate
This small group activity using surfaces introduces a geometric interpretation of partial derivatives in terms of measured ratios of small changes. Students work in small groups to identify locations on their surface with particular properties. The whole class wrap-up discussion emphasizes the equivalence of multiple representations of partial derivatives.
Find the total differential of the following functions:
  1. \(y=3u^2 + 4\cos 3v\)
  2. \(y=3uv\)
  3. \(y=3u^2\cos wv\)
  4. \(y=u\cos(3v^2-2)\)
  • Found in: Static Fields, AIMS Maxwell course(s)

group Small Group Activity

60 min.

The Park

This is the first activity relating the surfaces to the corresponding contour diagrams, thus emphasizing the use of multiple representations.

Students work in small groups to interpret level curves representing different concentrations of lead.

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.
  • Found in: Thermal and Statistical Physics course(s)