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Quantum states are traditionally cognitively managed exclusively with algebra rather than geometry. One reason for emphasizing algebra is the high dimensionality of quantum mathematical systems; even spin-1/2 systems require a 2-d complex number space for describing their quantum states, which can be hard to visualize. Using "nested phasor diagrams," which use nesting to increase the dimensionality of graphic space, we taught undergraduate students to represent spin-1/2 states graphically as well as algebraically. In oral exams, students were asked to identify which spin-1/2 states, expressed arithmetically, would generate the same set of probabilities as each other (i.e., they are the same except for a different overall phase factor). Video records of oral exams (N=13) show that no student performed this task successfully using an algebraic method; instead, all successful students solved the problem graphically. Furthermore, most students who were successful used a certain gesture to solve the problem.

Students' dynamic geometric reasoning about quantum spin-1/2 states