States

• First ask the students to show a particular spin 1/2 state, specified by a normalized pair of complex numbers. For example: \[\frac{1}{\sqrt{2}} \begin{pmatrix} 1\\e^{\frac{i\pi}{3}} \end{pmatrix}\]
• Next, ask the students to show ALL the states equivalent to this one. Each pair of students should rotate their arms simultaneously, by the same amount.
• Then, ask students to show any different state. Bring out the fact that the relative angle between the two students arms must now be different.
• 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.