Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.
1. << Time Evolution of a Quantum Particle on a Ring with Arms | Arms Sequence for Complex Numbers and Quantum States |
This activity is part of the Arms Sequence for Complex Numbers and Quantum States. If you have not used previous activities in the sequence, you may want to start with the introduction and a few of the prompts as listed in the first activity: Using Arms to Visualize Complex Numbers (MathBits) and a few of the prompts as listed in the third activity Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems.
This is a discovery activity that prepares students to understand the solution of the Schrodinger Equation.
Set-Up
Optional Warm-Up Have students represent the following states: \(\left|{+}\right\rangle _x\), \(\left|{-}\right\rangle _x\), \(\left|{+}\right\rangle _y\), \(\left|{-}\right\rangle _y\). This will help students recognize these states when they come up later in the activity.
“Represent the state \(\left|{+}\right\rangle _x\). Pick some arbitrary overall phase.”
“Represent the state \(e^{i\omega t}\left|{+}\right\rangle _x\). What does it look like? Does this state change with time?”
“What kind of motion with your arms would result in a state that changes with time?”
“How could you represent this time-evolving state in Dirac notation?”
Optional Concrete Example - Spin Precession: Ask to student to start in the state \(\left|{+}\right\rangle _x\) and act out the time evolution:
\[\left|{\psi(t)}\right\rangle = \frac{1}{\sqrt{2}}e^{i\omega\;t}\left|{+}\right\rangle +\frac{1}{\sqrt{2}}e^{-i\omega\;t}\left|{-}\right\rangle \]