Visualizing Complex Numbers

This sequence of activities introduces students to a kinesthetic way of representing complex-valued numbers and vectors. Students use their left arm as an Argand diagram to represent various complex numbers and coefficients of spin-1/2 systems. The sequence begins with individual students representing complex numbers and builds to pairs of students, representing the two-component complex-valued vectors of spin-1/2 systems. These activities provide 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 Numbers (Estimated time: 5 minutes): This kinesthetic activity introduces to students to the Argand diagram representation of complex numbers given in rectangular ($x+iy$) and exponential ($re^{i\phi}$) forms.
  • Visualizing Complex Time Dependence for Spin 1/2 System (Estimated time: 10-15 minutes): This kinesthetic activity is designed to help students visualize complex time dependence of the spin-1/2 system by using their arms, in pairs, to represent the different time dependencies of the eigenstates and the superposition states. The whole class discussion focuses on how different states vary with time.

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