“Arms” is an engaging representation of complex numbers in which 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 Guide for the second activity for ideas about how to introduce the Arms representation the first time you use it.

phase transformationClausius-ClapeyronThermal and Statistical Physics 2020
Consider a phase
transformation between either solid or liquid and gas. Assume that the
volume of the gas is way bigger than that of the liquid or
solid, such that \(\Delta V \approx V_g\). Furthermore, assume that
the ideal gas law applies to the gas phase. Note: this problem
is solved in the textbook, in the section on the Clausius-Clapeyron
equation.

Solve for \(\frac{dp}{dT}\) in terms of the pressure of the vapor
and the latent heat \(L\) and the temperature.

Assume further that the latent heat is roughly independent of
temperature. Integrate to find the vapor pressure itself as a
function of temperature (and of course, the latent heat).

Students, working in pairs, represent two component complex vectors with their left arms. Through a short series of instructor led prompts, students move their left arms to show how various linear transformations affect each complex component.

These lecture notes from the ninth week of Thermal and Statistical Physics cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.

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.