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“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.

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.

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.