None

##### Ring Table

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

• Found in: Central Forces course(s)

group Small Group Activity

10 min.

##### Using Tinker Toys to Represent Spin 1/2 Quantum Systems
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.

• Found in: Arms Sequence for Complex Numbers and Quantum States sequence(s)

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.

face Lecture

5 min.

##### Unit Learning Outcomes: Quantum Mechanics on a Ring
• Found in: Central Forces course(s)

None

##### Spin-1 Eigenvectors
The operator $\hat{S}_x$ for spin-1 may be written as: defined by: $\hat{S}_x=\frac{\hbar}{\sqrt{2}} \begin{pmatrix} 0&1&0\\ 1&0&1 \\ 0&1&0 \\ \end{pmatrix}$
1. Find the eigenvalues and eigenvectors of this matrix. Write the eigenvectors as both matrices and kets.
2. Confirm that the eigenstates you found give probabilities that match your expectation from the Spins simulation for spin-1 particles.
• Found in: Quantum Fundamentals course(s)

group Small Group Activity

30 min.

##### Expectation Values for a Particle on a Ring
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.

• Found in: Central Forces course(s) Found in: Quantum Ring Sequence sequence(s)

group Small Group Activity

30 min.

##### Superposition States for a Particle on a Ring
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.

• Found in: Quantum Ring Sequence sequence(s)

accessibility_new Kinesthetic

30 min.

##### Time Evolution of a Quantum Particle on a Ring with Arms
Students use their arms to act out stationary and non-stationary states of a quantum particle on a ring.

• Found in: Arms Sequence for Complex Numbers and Quantum States sequence(s)

None

##### Spin Three Halves Time Dependence
A spin-3/2 particle initially is in the state $|\psi(0)\rangle = |\frac{1}{2}\rangle$. This particle is placed in an external magnetic field so that the Hamiltonian is proportional to the $\hat{S}_x$ operator, $\hat{H} = \alpha \hat{S}_x \doteq \frac{\alpha\hbar}{2}\begin{pmatrix} 0 & \sqrt{3} & 0 & 0\\ \sqrt{3} & 0 & 2 & 0\\ 0 & 2 & 0 & \sqrt{3} \\ 0 & 0 & \sqrt{3} & 0 \end{pmatrix}$
1. Find the energy eigenvalues and energy eigenstates for the system.
2. Find $|\psi(t)\rangle$.
3. List the outcomes of all possible measurements of $S_x$ and find their probabilities. Explicitly identify any probabilities that depend on time.
4. List the outcomes of all possible measurements of $S_z$ and find their probabilities. Explicitly identify any probabilities that depend on time.
• Found in: Quantum Fundamentals course(s)

face Lecture

5 min.

##### Quantum Reference Sheet
• Found in: Central Forces course(s)

group Small Group Activity

120 min.

##### Spin-1 Time Evolution
Students do calculations for time evolution for spin-1.

• Found in: Quantum Fundamentals course(s)

None

##### Normalization of Quantum States
Show that if a linear combination of ring energy eigenstates is normalized, then the coefficients must satisfy $$\sum_{m=-\infty}^{\infty} \vert c_m\vert^2=1$$
• Found in: Central Forces course(s)

face Lecture

30 min.

##### Time Evolution Refresher (Mini-Lecture)
The instructor gives a brief lecture about time dependence of energy eigenstates (e.g. McIntyre, 3.1). Notes for the students are attached.

• Found in: Central Forces course(s) Found in: Quantum Ring Sequence sequence(s)

group Small Group Activity

30 min.

##### Time Evolution of a Spin-1/2 System
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.

• Found in: Quantum Fundamentals course(s)

None

##### ISW Position Measurement

A particle in an infinite square well potential has an initial state vector $\left|{\Psi(0)}\right\rangle = A\big(\left|{\phi_1}\right\rangle -\left|{\phi_2}\right\rangle +i\left|{\phi_3}\right\rangle \big)$

where $|\phi_n\rangle$ are the energy eigenstates. You have previously found $\left|{\Psi(t)}\right\rangle$ for this state.

1. Use a computer to graph the wave function $\Psi(x,t)$ and probability density $\rho(x,t)$. Choose a few interesting values of $t$ to include in your submission.

2. Use a computer to calculate the probability of measuring the particle to be near the middle of the well (within 1% on either side) as a function of time. Include both your symbolic result and a graph in your submission.

3. Choose another location in the well, different from the location above. Use a computer to calculate the probability of measuring the particle to be near your chosen location (within 1% on either side) as a function of time. Include both your symbolic result and a graph in your submission.

4. Are there any locations in the well where the probability is independent of time? Explain how you determined your answer.

5. The time dependence for a wave function like this is complicated. Write a lengthy description in words about the major features of this wave function and its probability density, how they change in time, and why they change the way they do. Comment on any interesting features you noticed that you have not already discussed in the questions above and describe any additional things you learned from the process of solving this problem.

• Found in: Quantum Fundamentals course(s)

None

##### Frequency
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.
• Found in: Quantum Fundamentals course(s)

group Small Group Activity

30 min.

##### Energy and Angular Momentum for a Quantum Particle on a Ring
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.

• Found in: Quantum Ring Sequence sequence(s)

group Small Group Activity

30 min.

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

• Found in: Quantum Fundamentals course(s)

None

##### Visualization of Wave Functions on a Ring
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.
• Found in: Central Forces course(s)

group Small Group Activity

30 min.

##### Working with Representations on the Ring
This activity acts as a reintroduction to doing quantum calculations while also introducing the matrix representation on the ring, allowing students to discover how to index and form a column vector representing the given quantum state. In addition, this activity introduces degenerate measurements on the quantum ring and examines the state after measuring both degenerate and non-degenerate eigenvalues for the state.

• Found in: Central Forces course(s)