Wave “Hello” to a New Way of Teaching Quantum Mechanics

Complex numbers play a fundamental role in quantum mechanics – and present a challenging aspect to teaching the fundamental theory. The increasingly popular spins-first approach introduces physics students to the subject by providing a conceptually rich introduction to the quantum postulates without requiring advanced mathematics needed to solve differential equations. The trade-off, however, is that students often contend with the complex nature of quantum states without fully grasping the significance and geometry of their complex vector components.

To help bridge the gap, many visualization options have been proposed, each with shortcomings that can convolute the already abstract concept of quantum states.

Hahn and Gire explained a novel “Arms representation” invented by their colleague Corinne Manogue as a tangible metaphor for quantum states, so other physics instructors may minimize abstraction and highlight key geometric relationships for students.

“We describe a series of kinesthetic instructional activities where students use their arms to represent quantum systems,” said author Kelby Hahn. “First, a student uses their physical arm to embody a complex number. Then in each subsequent activity, additional quantum mechanics ideas are layered on: spin states, relative and overall phase, time evolution, and wavefunctions.”

The Arms representation features students acting as Argand diagrams and using their left arms to represent numbers in a complex plane. This technique can be expanded to depict complex-valued vectors with groups of students.

“With the Arms representation, students feel and visualize some key quantum concepts,” said Hahn. “The activities target concepts students are known to struggle with, like the geometric meaning of relative and overall phase, or the transition from discrete to continuous observables.”

Source: “Waving arms around to teach quantum mechanics,” by Kelby T. Hahn and Elizabeth Gire, American Journal of Physics (2022). The article can be accessed at http://doi.org/10.1119/5.0073946 .