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## Visualizing Complex Time Dependence for Spin 1/2 System : Instructor's Guide

### Main Ideas

- Different forms of complex numbers ($x+iy$ and $re^{i\phi}$)
- Use of Argand diagrams
- Time evolution of states in spin 1/2 systems

### Students' Task

*Estimated Time: 10-15 minutes*

Students use their left arm as an Argand diagram and work in pairs to demonstrate eigenstates and superposition states of the spin-1/2 system with and without time dependence.

### Prerequisite Knowledge

### Props/Equipment

- Pairs of students

### Activity: Introduction

Students use the embodied experience of using their left arm as an Argand diagram where the left shoulder is the origin and practice using this to represent various spin-1/2 state. The same convention should be chosen across the classroom; typically the partner on the left side represents the coefficients associated with spin-up, and the one on the right side represents spin-down.

### Activity: Student Conversations

- Representing spin-1/2 systems at a particular time
- Relative & Overall Phase: At a particular time (such as $t=0$ in most time evolution problems), there is a fixed relative phase, however, the overall phase does not matter. Students may choose to represent the spin-1/2 state with various overall phases but while maintaining the same relative phase. It is important to point out to students that changing the overall phase does not affect the state of the system.

- Representing time evolved spin-1/2 systems
- Relative & Overall Phase: With time evolved states, the relative phase changes with time. Many students may first attempt to maintain a fixed relative phase before recognizing that the coefficients contain functions of time.
- Frequency: The complex function coefficients have different energies associated with them which give different frequencies in the complex plane associated with the two components. Students must consider the direction and period of the complex function associated with the ket they are representing and compare to their partner's in order to construct the time evolved representation.

- Using “tinker toys” with (or instead of) embodied experience
- Students can work in pairs or independently to equivalently represent these spin-1/2 states using tinker toys. This allows for differences in “arm” length on the kets but still allows for rotations. [: should this be a separate link? This is not explained in enough detail.]

### Activity: Wrap-up

This activity concludes with a whole class discussion on time evolution and can lead into calculations of probabilities associated with the time evolved state.

### Extensions

This activity is the final activity of the sequence addressing Visualizing Complex Numbers in the context of quantum mechanics.

- Preceding activities:
- Visualizing Complex Numbers: This kinesthetic activity introduces students to the rectangular and exponential forms of complex numbers.
- Visualizing Complex Two-Component Vectors: In this kinesthetic activity, students work in pairs to represent complex two-component vectors and transformations on those vectors.