This sequence of activities introduces students to a kinesthetic way of representing complex-valued vectors. Students use their left arm as an Argand diagram and then a series of students act as different components of a complex-valued vector. The sequence starts with pairs of students, representing the two-component, complex-valued vectors of spin 1/2 systems. These activities provide the students with a concrete way of visualizing both real and complex transformations, of distinguishing between a relative phase and an overall phase, and of grasping the differing time dependencies of the components of a superposition state.

• Visualizing Complex Two Component Vectors
  • This first activity introduces students to the representation and has them work in pairs to represent complex two component vectors and various types of transformations on those vectors. This follows naturally from the Linear Transformations Activity and the Eigenvectors and Eigenvalues Activity.
• Visualizing Complex Time Dependence for Spin 1/2 System
  • Description of activity