Spring Force Constant

• group A glass of water

  group Small Group Activity

  30 min.

  A glass of water

  Energy and Entropy 2021 (2 years)

  thermodynamics intensive extensive temperature volume energy entropy

  Students generate a list of properties a glass of water might have. The class then discusses and categorizes those properties.
• group Applying the equipartition theorem

  group Small Group Activity

  30 min.

  Applying the equipartition theorem

  Contemporary Challenges 2021 (4 years)

  equipartition theorem

  Students count the quadratic degrees of freedom of a few toy molecules to predict their internal energy at temperature \(T\).
• groups Pineapples and Pumpkins

  groups Whole Class Activity

  10 min.

  Pineapples and Pumpkins

  Static Fields 2023 (6 years)

  Integration Sequence

  There are two versions of this activity:

  As a whole class activity, the instructor cuts a pumpkin in order to produce a small volume element \(d\tau\), interspersing their work with a sequence of small whiteboard questions. This version of the activity is described here.

  As a small group activity, students are given pineapple rounds and pumpkin wedges to explore area volume elements in cylindrical and spherical coordinate systems. In this version of the activity, the fruit is distribued to the students with appropriate children's pumpkin cutting equipment, as part of activities Vector Differential--Curvilinear, Scalar Surface and Volume Elements, or Vector Surface and Volume Elements.

• face Equipartition theorem

  face Lecture

  30 min.

  Equipartition theorem

  Contemporary Challenges 2021 (4 years)

  equipartition heat capacity

  This lecture introduces the equipartition theorem.
• group Gaussian Parameters

  group Small Group Activity

  10 min.

  Gaussian Parameters

  Periodic Systems 2022

  Fourier Transforms and Wave Packets

  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}
• group Fourier Transform of a Shifted Function

  group Small Group Activity

  5 min.

  Fourier Transform of a Shifted Function

  Periodic Systems 2022

  Fourier Transforms and Wave Packets

• assignment Spin Three Halves Time Dependence

  assignment Homework

  Spin Three Halves Time Dependence

  Quantum Fundamentals 2023 A spin-3/2 particle initially is in the state \(|\psi(0)\rangle = |\frac{1}{2}\rangle\). This particle is placed in an external magnetic field so that the Hamiltonian is proportional to the \(\hat{S}_x\) operator, \(\hat{H} = \alpha \hat{S}_x \doteq \frac{\alpha\hbar}{2}\begin{pmatrix} 0 & \sqrt{3} & 0 & 0\\ \sqrt{3} & 0 & 2 & 0\\ 0 & 2 & 0 & \sqrt{3} \\ 0 & 0 & \sqrt{3} & 0 \end{pmatrix}\)

  1. Find the energy eigenvalues and energy eigenstates for the system.
  2. Find \(|\psi(t)\rangle\).
  3. List the outcomes of all possible measurements of \(S_x\) and find their probabilities. Explicitly identify any probabilities that depend on time.
  4. List the outcomes of all possible measurements of \(S_z\) and find their probabilities. Explicitly identify any probabilities that depend on time.
• assignment_ind Normalization of the Gaussian for Wavefunctions

  assignment_ind Small White Board Question

  5 min.

  Normalization of the Gaussian for Wavefunctions

  Periodic Systems 2022

  Fourier Transforms and Wave Packets

  Students find a wavefunction that corresponds to a Gaussian probability density.
• assignment Building the PDM: Instructions

  assignment Homework

  Building the PDM: Instructions

  PDM Energy and Entropy 2021 (2 years) In your kits for the Portable Partial Derivative Machine should be the following:

  • A 1ft by 1ft board with 5 holes and measuring tapes (the measuring tapes will be on the top side)
  • 2 S-hooks

    

  • A spring with 3 strings attached
  • 2 small cloth bags
  • 4 large ball bearings
  • 8 small ball bearings
  • 2 vertical clamp pulleys
  • A ziploc bag containing
    • 5 screws
    • 5 hex nuts
    • 5 washers
    • 5 wing nuts
    • 2 horizontal pulleys

  To assemble the Portable PDM, start by placing the PDM on a table surface with the measuring tapes perpendicular to the table's edge and the board edge with 3 holes closest to you.

  1. one screw should be put through each hole so that the threads stick out through the top side of the board. Next use a hex nut to secure each screw in place. It is not critical that they be screwed on any more than you can comfortably manage by hand.
  2. After securing all 5 screws in place with a hex nut, put a washer on each screw.
  3. Slide a horizontal pulley onto screws 1 and 2 (as labeled above).
  4. On all 5 screws, add a wing nut to secure the other pieces. Again, it does not need to be tightened all the way as long as it is secure enough that nothing will fall off.
  5. Using the middle wingnut/washer/screw (Screw 4), clamp the shortest of the strings tied to the spring.
  6. Loop the remaining 2 looped-ends of string around the horizontal pulleys and along the measuring tape.

     

  7. Using the string as a guide, clamp the vertical pulleys into place on the edge of the board.
  8. Through the looped-end of each string, place 1 S-hook.
  9. Put the other end of each s-hook through the hole in the small cloth bag.

  Here is a poor photo of the final result, which doesn't show the two vertical pulleys. If you would like, you could view a video of the building process.
• group Fourier Transform of the Delta Function

  group Small Group Activity

  5 min.

  Fourier Transform of the Delta Function

  Periodic Systems 2022

  Fourier Transforms and Wave Packets

  Students calculate the Fourier transform of the Dirac delta function.
• Energy and Entropy 2021 (2 years) The spring constant \(k\) for a one-dimensional spring is defined by: \[F=k(x-x_0).\] Discuss briefly whether each of the variables in this equation is extensive or intensive.