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.

Completeness Relations

This set of activities explores completeness relations in quantum mechanics and how they eventually support understanding of wavefunctions and what it means to be in the position or momentum representatations.

Curvilinear Coordinate Sequence

The curvilinear coordinate sequence introduces cylindrical and spherical coordinates (including inconsistencies between physicists' and mathematicians' notational conventions) and the basis vectors adapted to these coordinate systems.

E&M Ring Cycle Sequence

Students calculate electrostatic fields (\(V\), \(\vec{E}\)) and magnetostatic fields (\(\vec{A}\), \(\vec{B}\)) from charge and current sources with a common geometry. The sequence of activities is arranged so that the mathematical complexity of the formulas students encounter increases with each activity. Several auxiliary activities allow students to focus on the geometric/physical meaning of the distance formula, charge densities, and steady currents. A meta goal of the entire sequence is that students gain confidence in their ability to parse and manipulate complicated equations.

Fourier Transforms and Wave Packets

This is a unit that introduces the Fourier transform and its properties and then applies the Fourier transform to free particle wave packets in non-relativistic quantum mechanics. The activities and homework are listed here. Appropriate text materials for mini-lectures can be found in the chapter Fourier Transforms and Wave Packets in the free online textbook The Geometry of Mathematical Methods.

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.

Integration Sequence

Students learn/review how to do integrals in a multivariable context, using the vector differential \(d\vec{r}=dx\, \hat{x}+dy\, \hat{y}+dz\, \hat{z}\) and its curvilinear coordinate analogues as a unifying strategy. This strategy is common among physicists, but is NOT typically taught in vector calculus courses and will be new to most students.

Power Series Sequence (E&M)

The first three activities provide an active-engagement version of the canonical mathematical and geometric fundamentals for power series. The subsequent activities apply these ideas to physical situations that are appropriate for an upper-division electromagnetism course, using concepts, terminology, and techniques that are common among physicists, but not often taught in mathematics courses. In particular students use the memorized formula for the binomial expansion to evaluate various electrostatic and magnetostatic field in regions of high symmetry. By factoring out a physical quantity which is large compared to another physical quantity, they manipulate the formulas for these fields into a form where memorized formulas apply. The results for the different regions of high symmetry are compared and contrasted. A few homework problems that emphasize the meaning of series notation are included.

Note: The first two activities are also included in Power Series Sequence (Mechanics) and can be skipped in E&M if already taught in Mechanics.

Quantum Ring Sequence

Students calculate probabilities and expectation values for a quantum mechanical particle confined to a circular ring in bra/ket, matrix, and wave function representations and compare the different calculation methods. Several different graphical representations of the time dependence for both states with special symmetry and arbitrary states are explored in a Mathematica notebook. Compared to the analogous particle-in-a-box, this quantum system has a new feature---degenerate energy eigenstates.


Warm-Up (Welcome Activity Reviewing Material from Undergraduate Physics)

This content is used in the Physics Department at OSU with incoming graduate students to remind them of undergraduate content before classes start and to help them to decide whether or not to take some Bridge Courses. This sequence is intended to run in two blocks of three hours each. The sessions should be run by someone with a deep knowledge of all of the relevant courses, the specific activities, and active engagement in general.

This session may be the first opportunity for the incoming graduate students to meet each other as well as some faculty and other graduate students. So start with a 1/2 hour dedicated to introductons.

Consider inviting some or all of the following people to participate:

  • At least one faculty member to run the session who has broad experience with the curriculum and the activities--typically the Paradigms Director.
  • Graduate students who have TAd for courses that incorporated these exact activities, as needed to provide one experienced person to sit with each group of three graduate students. The Head Graduate Advisor has often asked these graduate students for evaluative input regarding the members of their group. CAM thinks that they should be given a heads-up about what will be expected.
  • The Head Graduate Advisor (n.b. In the past the Grad Advisor has roamed the classroom, hovering over the groups as they work. CAM thinks this can appear intimidating/judgmental. Consider asking the grad advisor to SIT with groups, even if they move frequently from group to group.
  • Members of the Core Advising Committee
  • Faculty who will be teaching the Bridge Courses so that they are available to answer student questions, especially individual questions during breaks.
  • Graduate students who have take Bridge Courses in the past who are comfortable discussing their choices and experiences.