Students use Tinker Toys to represent each component in a two-state quantum spin system in all three standard bases (\(x\), \(y\), and \(z\)). Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT change the state of the system) and relative phase (which does change the state of the system). This activity is optional in the Arms Sequence Arms Sequence for Complex Numbers and Quantum States.
1. << Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems | Arms Sequence for Complex Numbers and Quantum States | Using Arms to Represent Time Dependence in Spin 1/2 Systems >>
Students should already have some experience with the \(S_x\), \(S_y\), and \(S_z\) eigenstates for a spin 1/2 system, all written in the \(z\) basis.
Draw the complex plane on a board at the front of the room and provide a list of the \(S_x\), \(S_y\), and \(S_z\) eigenstates for a spin 1/2 system, all written in the \(z\) basis.
\[ \begin{align*} S_z: \qquad\qquad \begin{pmatrix} 1\\0 \end{pmatrix} \qquad &\begin{pmatrix} 0\\1 \end{pmatrix}\\ S_x: \qquad \frac{1}{\sqrt{2}}\begin{pmatrix} 1\\1 \end{pmatrix} \qquad &\frac{1}{\sqrt{2}}\begin{pmatrix} 1\\-1 \end{pmatrix}\\ S_y: \qquad \frac{1}{\sqrt{2}}\begin{pmatrix} 1\\i \end{pmatrix} \qquad &\frac{1}{\sqrt{2}}\begin{pmatrix} 1\\-i \end{pmatrix}\\ \end{align*} \]
Using a center piece (for the origin) and a long straight piece, demonstrate how a complex number can be represented with Tinker Toys.
Now ask the student groups to connect two center pieces with a short connector through their centers. Then have them build a representation of the \(S_x\), \(S_y\), and \(S_z\) eigenstates.
Here is an image of the three sets of eigenstates:
The complex numbers are in the standard orientation (positive real axis to the right).
The short (green) connector has no physical/geometric meaning.
Discuss how you can tell that each model represents a different state: i.e. they all have a different relative phase between the two complex numbers.
Discuss how the models can represent the overall phase independence of the state: i.e. any rotation of the model around its vertical axis represents the same state.
accessibility_new Kinesthetic
10 min.
quantum states complex numbers arms Bloch sphere relative phase overall phase
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).group Small Group Activity
30 min.
group Small Group Activity
30 min.
group Small Group Activity
10 min.
group Small Group Activity
30 min.
assignment Homework
assignment Homework
Writing an operator in matrix notation in its own basis is easy: it is diagonal with the eigenvalues on the diagonal.
What if I want to calculate the matrix elements using a different basis??
The eigenvalue equation tells me what happens when an operator acts on its own eigenstate. For example: \(\hat{S}_y\left|{\pm}\right\rangle _y=\pm\frac{\hbar}{2}\left|{\pm}\right\rangle _y\)
In Dirac bra-ket notation, to know what an operator does to a ket, I needs to write the ket in the basis that is the eigenstates of the operator (in order to use the eigenvalue equation.)
One way to do this to stick completeness relationships into the braket: \begin{eqnarray*} \left\langle {+}\right|\hat{S_y}\left|{+}\right\rangle = \left\langle {+}\right|(I)\hat{S_y}(I)\left|{+}\right\rangle \end{eqnarray*}
where \(I\) is the identity operator: \(I=\color{blue}{\left|{+}\right\rangle _{yy}\left\langle {+}\right|}\;+\;\color{blue}{\left|{-}\right\rangle _{yy}\left\langle {-}\right|}\). This effectively rewrite the \(\left|{+}\right\rangle \) in the \(\left|{\pm}\right\rangle _y\) basis.
Find the top row matrix elements of the operator \(\hat{S}_y\) in the \(S_z\) basis by inserting completeness relations into the brakets. (The answer is already on the Spins Reference Sheet, but I want you do demonstrate the calculation.)
group Small Group Activity
60 min.
assignment Homework
group Small Group Activity
30 min.