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 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.

group Small Group Activity

10 min.

Using Tinker Toys to Represent Spin 1/2 Quantum Systems

spin 1/2 eigenstates quantum states

Arms Sequence for Complex Numbers and Quantum States

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.

group Small Group Activity

30 min.

Expectation Values for a Particle on a Ring
Central Forces 2023 (3 years)

central forces quantum mechanics eigenstates eigenvalues hermitian operators quantum measurements degeneracy expectation values time dependence

Quantum Ring Sequence

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.

group Small Group Activity

30 min.

Superposition States for a Particle on a Ring

central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum hermitian operators probability superposition

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.

face Lecture

5 min.

Quantum Reference Sheet
Central Forces 2023 (6 years)

assignment Homework

Normalization of Quantum States
Central Forces 2023 (3 years) Show that if a linear combination of ring energy eigenstates is normalized, then the coefficients must satisfy \begin{equation} \sum_{m=-\infty}^{\infty} \vert c_m\vert^2=1 \end{equation}

face Lecture

30 min.

Time Evolution Refresher (Mini-Lecture)
Central Forces 2023 (3 years)

schrodinger equation time dependence stationary states

Quantum Ring Sequence

The instructor gives a brief lecture about time dependence of energy eigenstates (e.g. McIntyre, 3.1). Notes for the students are attached.

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 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 \begin{equation} \hat{H}\doteq \begin{pmatrix} E_1&0\\ 0&E_2 \end{pmatrix} \end{equation} Another physical observable \(M\) is described by the operator \begin{equation} \hat{M}\doteq \begin{pmatrix} 0&c\\ c&0 \end{pmatrix} \end{equation} 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 Small Group Activity

30 min.

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

central forces quantum mechanics eigenstates eigenvalues angular momentum time dependence hermitian operators probability degeneracy quantum measurements

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.

group Small Group Activity

30 min.

Energy and Angular Momentum for a Quantum Particle on a Ring

central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum energy hermitian operators probability superposition representations notations degeneracy

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 Small Group Activity

30 min.

Quantum Measurement Play
Quantum Fundamentals 2022 (2 years)

Quantum Measurement Projection Operators Spin-1/2

The instructor and students do a skit where students represent quantum states that are “measured” by the instructor resulting in a state collapse.

assignment Homework

Visualization of Wave Functions on a Ring
Central Forces 2023 (3 years) Using either this Geogebra applet or this Mathematica notebook, explore the wave functions on a ring. (Note: The Geogebra applet may be a little easier to use and understand and is accessible if you don't have access to Mathematica, but it is more limited in the wave functions that you can represent. Also, the animation is pretty jumpy in some browsers, especially Firefox. Imagine that the motion is smooth.)
  1. Look at graphs of the following states \begin{align} \Phi_1(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle +\left|{-2}\right\rangle )\\ \Phi_2(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle -\left|{-2}\right\rangle )\\ \Phi_3(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle +i\left|{-2}\right\rangle ) \end{align} Write a short description of how these states differ from each other.
  2. Find a state for which the probability density does not depend on time. Write the state in both ket and wave function notation. These are called stationary states. Generalize your result to give a characterization of the set of all possible states that are stationary states.
  3. Find a state that is right-moving. Write the state in both ket and wave function notation. Generalize your result to give a characterization of the set of all possible states that are right-moving.
  4. Find a state that is a standing wave. Write the state in both ket and wave function notation. Generalize your result to give a characterization of the set of all possible states that are standing waves.

assignment Homework

Quantum Particle in a 2-D Box
Central Forces 2023 (3 years) You know that the normalized spatial eigenfunctions for a particle in a 1-D box of length \(L\) are \(\sqrt{\frac{2}{L}}\sin{\frac{n\pi x}{L}}\). If you want the eigenfunctions for a particle in a 2-D box, then you just multiply together the eigenfunctions for a 1-D box in each direction. (This is what the separation of variables procedure tells you to do.)
  1. Find the normalized eigenfunctions for a particle in a 2-D box with sides of length \(L_x\) in the \(x\)-direction and length \(L_y\) in the \(y\)-direction.
  2. Find the Hamiltonian for a 2-D box and show that your eigenstates are indeed eigenstates and find a formula for the possible energies
  3. Any sufficiently smooth spatial wave function inside a 2-D box can be expanded in a double sum of the product wave functions, i.e. \begin{equation} \psi(x,y)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\; \hbox{eigenfunction}_n(x)\;\hbox{eigenfunction}_m(y) \end{equation} Using your expressions from part (a) above, write out all the terms in this sum out to \(n=3\), \(m=3\). Arrange the terms, conventionally, in terms of increasing energy.

    You may find it easier to work in bra/ket notation: \begin{align*} \left|{\psi}\right\rangle &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{n}\right\rangle \left|{m}\right\rangle \\ &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{nm}\right\rangle \end{align*}

  4. Find a formula for the \(c_{nm}\)s in part (b). Find the formula first in bra ket notation and then rewrite it in wave function notation.

group Small Group Activity

30 min.

Hydrogen emission
Contemporary Challenges 2022 (5 years)

hydrogen atom photon energy

In this activity students work out energy level transitions in hydrogen that lead to visible light.

computer Mathematica Activity

30 min.

Visualization of Quantum Probabilities for a Particle Confined to a Ring
Central Forces 2023 (3 years)

central forces quantum mechanics angular momentum probability density eigenstates time evolution superposition mathematica

Quantum Ring Sequence

Students see probability density for eigenstates and linear combinations of eigenstates for a particle on a ring. The three visual representations: standard position vs probability density plot, a ring with colormapping, and cylindrical plot with height and colormapping, are also animated to visualize time-evolution.

computer Mathematica Activity

30 min.

Visualizing Combinations of Spherical Harmonics
Central Forces 2023 (3 years) Students observe three different plots of linear combinations of spherical combinations with probability density represented by color on the sphere, distance from the origin (polar plot), and distance from the surface of the sphere.

group Small Group Activity

120 min.

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

Warm-Up

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.