Students, working in pairs, use the Arms representations to represent states of spin 1/2 system. Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT distinguish quantum states) and relative phase (which does distinguish quantum states).
1. << Inner Product of Spin-1/2 System with Arms | Arms Sequence for Complex Numbers and Quantum States | Using Tinker Toys to Represent Spin 1/2 Quantum Systems >>
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).
Give a brief introductory lecture:
Have the students pair up and write a state on the board.
Have them close their eyes and act out the given pair of complex numbers.
Complex numbers can be given in both rectangular \(x+iy\) and exponential \(re^{i\phi}\) forms.
After each prompt, have students open their eyes and compare. Discuss as necessary.
If your students have already learned how to represent the states that are spin up or down in the \(z\), \(x\), and \(y\) directions, all represented in the \(z\)-basis, i.e.
\begin{align*} S_z: \qquad\qquad \begin{pmatrix} 1\\0 \end{pmatrix} \qquad &\begin{pmatrix} 0\\1 \end{pmatrix}\\ S_x: \qquad \frac{1}{\sqrt{2}}\begin{pmatrix} 1\\1 \end{pmatrix} \qquad &\frac{1}{\sqrt{2}}\begin{pmatrix} 1\\-1 \end{pmatrix}\\ S_y: \qquad \frac{1}{\sqrt{2}}\begin{pmatrix} 1\\i \end{pmatrix} \qquad &\frac{1}{\sqrt{2}}\begin{pmatrix} 1\\-i \end{pmatrix}\\ \end{align*}
then, have them represent each of these states with their arms.
Emphasize that it is the relative phase of a quantum state distinguishes different states. Two vectors that are only different by an overall phase describe the same quantum state.