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.

group Small Group Activity

5 min.

Fourier Transform of a Shifted Function
  • Found in: Periodic Systems course(s) Found in: Fourier Transforms and Wave Packets sequence(s)

None

Vapor pressure equation
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.

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

  2. 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).

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

group Small Group Activity

5 min.

Fourier Transform of the Delta Function
Students calculate the Fourier transform of the Dirac delta function.
  • Found in: Periodic Systems course(s) Found in: Fourier Transforms and Wave Packets sequence(s)

group Small Group Activity

5 min.

Fourier Transform of a Plane Wave
  • Found in: Periodic Systems course(s) Found in: Fourier Transforms and Wave Packets sequence(s)

accessibility_new Kinesthetic

30 min.

Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)
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.

face Lecture

120 min.

Phase transformations
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.

face Lecture

120 min.

Fermi and Bose gases
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.