Expectation Values and UncertaintyYou have a system that consists of quantum particles with spin. On this system, you will perform a Stern-Gerlach experiment with an analyzer oriented in the \(z\)-direction.
Consider one of the different initial spin states described below:
A spin 1/2 particle described by:
- \(\left|{+}\right\rangle \)
- \(\frac{i}{2}\left|{+}\right\rangle -\frac{\sqrt{3}}{2}\left|{-}\right\rangle \)
\(\left|{+}\right\rangle _x\)
A spin 1 particle described by:
- \(\left|{0}\right\rangle \)
- \(\left|{-1}\right\rangle _x\)
- \(\frac{2}{3}\left|{1}\right\rangle +\frac{i}{3}\left|{0}\right\rangle -\frac{2}{3}\left|{-1}\right\rangle \)
List the possible values of spin you could measure and determine the probability associated with each value of the z-component of spin.
Plot a histogram of the probabilities.
Find the expectation value of the z-component of spin.
- Find the uncertainty of the z-component of spin.
I like to break this activity into two parts:
(1) Calculating expectation values and relating them to the associated distributions of the probabilities of results, and
(2) Calculating the quantum uncertainty of the state and relating the uncertainty to distributions of the probabilities of results.
Therefore, I have my students do the first part of the activity before I introduce quantum uncertainty.
I introduce the activity by reminding students about two ways of calculating the expectation value. Given a quantum state \(\left|{\psi}\right\rangle \), for a measurement of an observable represented by an operator \(\hat{A}\) with eigenstates \(\left|{a_i}\right\rangle \) and eigenvalues \(a_i\): \begin{align*} \langle \hat{A} \rangle &= \sum_{i} a_i\mathcal{P}(a_i) \\ &= \left\langle {\psi}\right|\hat{A}\left|{\psi}\right\rangle \end{align*}
After the students calculate expectation values and we have a whole class discussion about 1 of the examples, then I do a lecture introducing the quantity of quantum uncertainty (relating it to the standard deviation of the distribution of probabilities by spin component value) and deriving the simplified equation:
\[\Delta A = \sqrt{\langle A^2 \rangle - \langle A \rangle ^2}\]
One could have each group report out, or the instructor could discuss a few key examples.
For expectation value, I like to talk about Case 2: \(\frac{i}{2}\left|{+}\right\rangle -\frac{\sqrt{3}}{2}\left|{-}\right\rangle \), where the probabilities of the two outcomes are not equal to show how the weighting plays out. Also, the expectation value is not a possible measurement value, and I like to talk about that. “Expectation” value is a misleading name for this quantity - it characterizes the distribution and is not necessarily a result of an individual measurement.
I also like to discuss an example like Case 5: \(\left|{1}\right\rangle _x\) where the distribution is symmetric around \(0\hbar\).
I think it's important to encourage students to calculate expectation values both ways (with probabilities and as a bracket with matrix notation) while the teaching team is available to help them.
For quantum uncertainty, I like to talk about an example like Case 3: \(\left|{+}\right\rangle _x\) where all the individual measurements are the same ”distance” away from the expectation value as a sensemaking exercise to connect to a conceptual interpretation of physics.
I also like to discuss an example like Case 5: \(\left|{-1}\right\rangle _x\), where the fact that we're taking an rms average is apparent: half the measurements are \(\hbar\) away from the expectation value and the other half are \(0\hbar\) away, but the uncertainty is \(\hbar/\sqrt{2}\).
assignment Homework
group Small Group Activity
30 min.
assignment Homework
group Small Group Activity
10 min.
group Small Group Activity
30 min.
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
30 min.
group Small Group Activity
120 min.
assignment Homework