title, topic, keyword
Small group, whiteboard, etc
Required in-class time for activities
Leave blank to search both

Activities

The Gibbs free energy, \(G\), is given by \begin{align*} G = U + pV - TS. \end{align*}
  1. Find the total differential of \(G\). As always, show your work.
  2. Interpret the coefficients of the total differential \(dG\) in order to find a derivative expression for the entropy \(S\).
  3. From the total differential \(dG\), obtain a different thermodynamic derivative that is equal to \[ \left(\frac{\partial {S}}{\partial {p}}\right)_{T} \]
Students practice using inner products to find the components of the cartesian basis vectors in the polar basis and vice versa. Then, students use a completeness relation to change bases or cartesian/polar bases and for different spin bases.

Problem

5 min.

Electric Field and Charge
Consider the electric field \begin{equation} \vec E(r,\theta,\phi) = \begin{cases} 0&\textrm{for } r<a\\ \frac{1}{4\pi\epsilon_0} \,\frac{Q}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ 0 & \textrm{for } r>b \\ \end{cases} \end{equation}
  1. (4pts) Use step and/or delta functions to write this electric field as a single expression valid everywhere in space.
  2. (4pts) Find a formula for the charge density that creates this electric field.
  3. (2pts) Interpret your formula for the charge density, i.e. explain briefly in words where the charge is.
Students work in small groups to use completeness relations to change the basis of quantum states.

Small Group Activity

30 min.

Completeness Relations
Students use a completeness relations to write hydrogen atoms states in the energy and position bases.
Students use completeness relations to write a matrix element of a spin component in a different basis.

Kinesthetic

10 min.

Curvilinear Basis Vectors
Students use their arms to depict (sequentially) the different cylindrical and spherical basis vectors at the location of their shoulder (seen in relation to a specified origin of coordinates: either a set of axes hung from the ceiling of the room or perhaps a piece of furniture or a particular corner of the room).
  • symmetry curvilinear coordinate systems basis vectors
    Found in: Static Fields, Central Forces, AIMS Maxwell, Surfaces/Bridge Workshop, Problem-Solving, None course(s) Found in: Geometry of Vector Fields Sequence, Curvilinear Coordinate Sequence sequence(s)

Small Group Activity

30 min.

Outer Product of a Vector on Itself
Students compute the outer product of a vector on itself to product a projection operator. Students discover that projection operators are idempotent (square to themselves) and that a complete set of outer products of an orthonormal basis is the identity (a completeness relation).

Small White Board Question

30 min.

Magnetic Moment & Stern-Gerlach Experiments
Students consider the relation (1) between the angular momentum and magnetic moment for a current loop and (2) the force on a magnetic moment in an inhomogeneous magnetic field. Students make a (classical) prediction of the outcome of a Stern-Gerlach experiment.

Lecture

120 min.

Phase transformations
These lecture notes from the ninth week of https://paradigms.oregonstate.edu/courses/ph441 cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.
Students use the completeness relation for the position basis to re-express expressions in bra/ket notation in wavefunction notation.