Activity: Quantum Expectation Values

Quantum Fundamentals 2022 (3 years)

Expectation Values and Uncertainty

You 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:

  1. \(\left|{+}\right\rangle \)
  2. \(\frac{i}{2}\left|{+}\right\rangle -\frac{\sqrt{3}}{2}\left|{-}\right\rangle \)
  3. \(\left|{+}\right\rangle _x\)

    A spin 1 particle described by:

  4. \(\left|{0}\right\rangle \)
  5. \(\left|{-1}\right\rangle _x\)
  6. \(\frac{2}{3}\left|{1}\right\rangle +\frac{i}{3}\left|{0}\right\rangle -\frac{2}{3}\left|{-1}\right\rangle \)
  • 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.

Introduction

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:

\[\Delta A = \sqrt{\langle A^2 \rangle - \langle A \rangle ^2}\]

Student Conversations

  1. 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 \(0\hbar\).

  2. 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.

  3. 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 Unknowns Spin-1/2 Brief

    assignment Homework

    Unknowns Spin-1/2 Brief
    Quantum Fundamentals 2022 (2 years) With the Spins simulation set for a spin 1/2 system, measure the probabilities of all the possible spin components for each of the unknown initial states \(\left|{\psi_3}\right\rangle \) and \(\left|{\psi_4}\right\rangle \).
    1. Use your measured probabilities to find each of the unknown states as a linear superposition of the \(S_z\)-basis states \(\left|{+}\right\rangle \) and \(\left|{-}\right\rangle \).
    2. Articulate a Process: Write a set of general instructions that would allow another student in next year's class to find an unknown state from measured probabilities.
    3. Compare Theory with Experiment: Design an experiment that will allow you to test whether your prediction for each of the unknown states is correct. Describe your experiment here, clearly but succinctly, as if you were writing it up for a paper. Do the experiment and discuss your results.
    4. Make a Conceptual Connection: In general, can you determine a quantum state with spin-component probability measurements in only two spin-component-directions? Why or why not?
  • group Time Evolution of a Spin-1/2 System

    group Small Group Activity

    30 min.

    Time Evolution of a Spin-1/2 System
    Quantum Fundamentals 2022 (3 years)

    quantum mechanics spin precession time evolution

    In this small group activity, students solve for the time dependence of two quantum spin 1/2 particles under the influence of a Hamiltonian. Students determine, given a Hamiltonian, which states are stationary and under what circumstances measurement probabilities do change with time.
  • assignment Phase 2

    assignment Homework

    Phase 2
    quantum mechanics relative phase overall phase measurement probability Quantum Fundamentals 2022 (2 years) Consider the three quantum states: \[\left\vert \psi_1\right\rangle = \frac{4}{5}\left\vert +\right\rangle+ i\frac{3}{5} \left\vert -\right\rangle\] \[\left\vert \psi_2\right\rangle = \frac{4}{5}\left\vert +\right\rangle- i\frac{3}{5} \left\vert -\right\rangle\] \[\left\vert \psi_3\right\rangle = -\frac{4}{5}\left\vert +\right\rangle+ i\frac{3}{5} \left\vert -\right\rangle\]
    1. For each of the \(\left|{\psi_i}\right\rangle \) above, calculate the probabilities of spin component measurements along the \(x\), \(y\), and \(z\)-axes.
    2. Look For a Pattern (and Generalize): Use your results from \((a)\) to comment on the importance of the overall phase and of the relative phases of the quantum state vector.
  • group Matrix Representation of Angular Momentum

    group Small Group Activity

    10 min.

    Matrix Representation of Angular Momentum
    Central Forces 2023 (2 years)
  • group Operators & Functions

    group Small Group Activity

    30 min.

    Operators & Functions
    Quantum Fundamentals 2022 (3 years) Students are asked to:
    • Test to see if one of the given functions is an eigenfunction of the given operator
    • See if they can write the functions that are found not to be eigenfunctions as a linear combination of eigenfunctions.
  • assignment Matrix Elements and Completeness Relations

    assignment Homework

    Matrix Elements and Completeness Relations
    Quantum Fundamentals 2022 (2 years)

    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 Finding if $S_{x}, \; S_{y}, \; and \; S_{z}$ Commute

    group Small Group Activity

    30 min.

    Finding if \(S_{x}, \; S_{y}, \; and \; S_{z}\) Commute
    Quantum Fundamentals 2022 (3 years)
  • group Representations of the Infinite Square Well

    group Small Group Activity

    120 min.

    Representations of the Infinite Square Well
    Quantum Fundamentals 2022 (3 years)

    Warm-Up

  • assignment Spin Fermi Estimate

    assignment Homework

    Spin Fermi Estimate
    Quantum Fundamentals 2022 The following two problems ask you to make Fermi estimates. In a good Fermi estimate, you start from basic scientific facts you already know or quantities that you can reasonably estimate based on your life experiences and then reason your way to estimate a quantity that you would not be able guess. You may look up useful conversion factors or constants. Use words, pictures, and equations to explain your reasoning:
    1. Imagine that you send a pea-sized bead of silver through a Stern-Gerlach device oriented to measure the z-component of intrinsic spin. Estimate the total z-component of the intrinsic spin of the ball you would measure in the HIGHLY improbable case that every atom is spin up.
    2. Protons, neutrons, and electrons are all spin-1/2 particles. Give a (very crude) order of magnitude estimate of the number of these particles in your body.
  • face Quantum Reference Sheet

    face Lecture

    5 min.

    Quantum Reference Sheet
    Central Forces 2023 (6 years)

Learning Outcomes