Activity: Expectation Values for a Particle on a Ring

Central Forces 2023 (3 years)
Students calculate the expectation value of energy and angular momentum as a function of time for an initial state for a particle on a ring. This state is a linear combination of energy/angular momentum eigenstates written in bra-ket notation.
• This activity is used in the following sequences
What students learn
• How to calculate expectation values
• Linear combinations of eigenstates with different eigenvalues are not stationary (yield expectation values that depend on time)
• Media

Expectation Values for a Particle on a Ring: Instructor's Guide

Expectation Values for a Particle on a Ring Handout

Introduction

It could be beneficial to see what students remember about expectation values by having asking them a Small White Board Question about the topic. Use their responses to drive an intro discussion about what expectation value is: not a measurement, but a weighted average.

Student Conversations

• Operators vs Measurement: Students commonly attempt to determine the values resulting from a quantum experiment by allowing the operator corresponding to the observable of interest to act on the initial state. Students who do this should be encouraged to consider the nature of this transformation (it's a vector, not a scalar) and to recognize that the transformation does not necessarily yield an eigenvector (the state of the system should be an eigenstate after the measurement).
• Degeneracy: Students may experience some difficulty due to the degeneracy of some states, in particular, that you have to include all the states that share that eigenvalue. $P_{E={m^2\,\hbar^2\over 2I}}=\vert \langle m\vert \psi\rangle\vert^2+\vert \langle -m\vert \psi\rangle\vert^2$
• Notation: As with earlier activities, students are usually comfortable at this point with doing expectation values in bra-ket notation, but fewer are comfortable with using wavefunction notation. For groups that finish early, ask them to use the other method to compare.
• Time Dependence: Some students might wonder about the time-(in)dependence of their expectation values. This is a good opportunity to remind students of Time Dependence for a Quantum Particle on a Ring where we discussed that quantities whose opperator commutes with the Hamiltonian will have time independent probabilities.

Wrap-up

This activity provides an opportunity to contrast two methods of finding expectation values.

• Carry out the explicit and messy differentiation and integration on the given state.
• Recast the initial state as a linear combination of eigenstates and carry out the much simpler calculations on these eigenstates.
Generally, students in the class will be mixed in the approach they choose. By emphasizing this when you wrapup this activity, students have the opportunity to sort out for themselves the benefits of each method. Remind them of Energy and Angular Momentum for a Quantum Particle on a Ring where they made this sort of comparison explicitly.

Discuss some sense-making techniques for expectation value such as: units of the expectation value, what the expectation values tells you about the distribution of possible measurement values. Drawing a histogram of possible measurements vs probability might be a good way to illustrate these properties to students.

Extensions and Related Materials

This is a part of Quantum Ring Sequence of activities.
• group Energy and Angular Momentum for a Quantum Particle on a Ring

group Small Group Activity

30 min.

Energy and Angular Momentum for a Quantum Particle on a Ring

Quantum Ring Sequence

Students calculate probabilities for a particle on a ring using three different notations: Dirac bra-ket, matrix, and wave function. After calculating the angular momentum and energy measurement probabilities, students compare their calculation methods for notation.
• group Time Dependence for a Quantum Particle on a Ring

group Small Group Activity

30 min.

Time Dependence for a Quantum Particle on a Ring
Theoretical Mechanics (6 years)

Quantum Ring Sequence

Students calculate probabilities for energy, angular momentum, and position as a function of time for an initial state that is a linear combination of energy/angular momentum eigenstates for a particle confined to a ring written in bra-ket notation. This activity helps students build an understanding of when they can expect a quantity to depend on time and to give them more practice moving between representations.
• assignment Ring Table

assignment Homework

Ring Table
Central Forces 2023 (3 years)

Attached, you will find a table showing different representations of physical quantities associated with a quantum particle confined to a ring. Fill in all of the missing entries. Hint: You may look ahead. We filled out a number of the entries throughout the table to give you hints about what the forms of the other entries might be. pdf link for the Table or doc link for the Table

• format_list_numbered Quantum Ring Sequence

format_list_numbered Sequence

Quantum Ring Sequence
Students calculate probabilities and expectation values for a quantum mechanical particle confined to a circular ring in bra/ket, matrix, and wave function representations and compare the different calculation methods. Several different graphical representations of the time dependence for both states with special symmetry and arbitrary states are explored in a Mathematica notebook. Compared to the analogous particle-in-a-box, this quantum system has a new feature---degenerate energy eigenstates.
• assignment Frequency

assignment Homework

Frequency
Quantum Mechanics Time Evolution Spin Precession Expectation Value Bohr Frequency Quantum Fundamentals 2022 (2 years) Consider a two-state quantum system with a Hamiltonian $$\hat{H}\doteq \begin{pmatrix} E_1&0\\ 0&E_2 \end{pmatrix}$$ Another physical observable $M$ is described by the operator $$\hat{M}\doteq \begin{pmatrix} 0&c\\ c&0 \end{pmatrix}$$ where $c$ is real and positive. Let the initial state of the system be $\left|{\psi(0)}\right\rangle =\left|{m_1}\right\rangle$, where $\left|{m_1}\right\rangle$ is the eigenstate corresponding to the larger of the two possible eigenvalues of $\hat{M}$. What is the frequency of oscillation of the expectation value of $M$? This frequency is the Bohr frequency.
• group Superposition States for a Particle on a Ring

group Small Group Activity

30 min.

Superposition States for a Particle on a Ring

Quantum Ring Sequence

Students calculate probabilities for a particle on a ring whose wavefunction is not easily separated into eigenstates by inspection. To find the energy, angular momentum, and position probabilities, students perform integrations with the wavefunction or decompose the wavefunction into a superposition of eigenfunctions.
• group Quantum Expectation Values

group Small Group Activity

30 min.

Quantum Expectation Values
Quantum Fundamentals 2022 (3 years)
• group Wavefunctions on a Quantum Ring

group Small Group Activity

30 min.

Wavefunctions on a Quantum Ring
Central Forces 2023 (2 years)
• group Expectation Value and Uncertainty for the Difference of Dice

group Small Group Activity

60 min.

Expectation Value and Uncertainty for the Difference of Dice
Quantum Fundamentals 2022 (3 years)
• assignment Energy of a relativistic Fermi gas

assignment Homework

Energy of a relativistic Fermi gas
Fermi gas Relativity Thermal and Statistical Physics 2020

For electrons with an energy $\varepsilon\gg mc^2$, where $m$ is the mass of the electron, the energy is given by $\varepsilon\approx pc$ where $p$ is the momentum. For electrons in a cube of volume $V=L^3$ the momentum takes the same values as for a non-relativistic particle in a box.

1. Show that in this extreme relativistic limit the Fermi energy of a gas of $N$ electrons is given by \begin{align} \varepsilon_F &= \hbar\pi c\left(\frac{3n}{\pi}\right)^{\frac13} \end{align} where $n\equiv \frac{N}{V}$ is the number density.

2. Show that the total energy of the ground state of the gas is \begin{align} U_0 &= \frac34 N\varepsilon_F \end{align}

Author Information
Corinne Manogue, Kerry Browne, Elizabeth Gire, Mary Bridget Kustusch, David McIntyre
Keywords
central forces quantum mechanics eigenstates eigenvalues hermitian operators quantum measurements degeneracy expectation values time dependence
Learning Outcomes