## Activity: Working with Representations on the Ring

Central Forces Spring 2021
• 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.
• assignment Ring Function

assignment Homework

##### Ring Function
Central Forces Spring 2021 Consider the normalized wavefunction $\Phi\left(\phi\right)$ for a quantum mechanical particle of mass $\mu$ constrained to move on a circle of radius $r_0$, given by: $$\Phi\left(\phi\right)= \frac{N}{2+\cos(3\phi)}$$ where $N$ is the normalization constant.
1. Find $N$.

2. Plot this wave function.
3. Plot the probability density.
4. Find the probability that if you measured $L_z$ you would get $3\hbar$.
5. What is the expectation value of $L_z$ in this state?
• accessibility_new Spin 1/2 with Arms

accessibility_new Kinesthetic

10 min.

##### Spin 1/2 with Arms

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, use their left arms to represent each component in a two-state quantum spin 1/2 system. Reinforces the idea that quantum states are complex valued vectors. Students make connections between Dirac, matrix, and Arms representation.
• 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.
• 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.
• 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
Central Forces Spring 2021

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 Going from Spin States to Wavefunctions

group Small Group Activity

60 min.

##### Going from Spin States to Wavefunctions

Arms Sequence for Complex Numbers and Quantum States

Students review using the Arms representation to represent states for discrete quantum systems and connecting the Arms representation to histogram and matrix representation. The student then extend the Arms representation to begin exploring the continuous position basis.
• assignment_ind Dot Product Review

assignment_ind Small White Board Question

10 min.

##### Dot Product Review
AIMS Maxwell AIMS 21 Static Fields Winter 2021

This small whiteboard question (SWBQ) serves as a quick review of the dot product. It is also an opportunity to help students see the advantages of knowing many different representations of and facts about a physical concept.
• assignment Lines in Polar Coordinates

assignment Homework

##### Lines in Polar Coordinates
Central Forces Spring 2021

The general equation for a straight line in polar coordinates is given by: $$r(\phi)=\frac{r_0}{\cos(\phi-\delta)}$$ Find the polar equation for the straight lines below. You do NOT need to evaluate any complicated trig or inverse trig functions. You may want to try plotting the general polar equation to figure out the roles of the parameters.

1. $y=3$
2. $x=3$
3. $y=-3x+2$

• assignment Series Convergence

assignment Homework

##### Series Convergence

Power Series Sequence (E&M)

AIMS Maxwell AIMS 21 Static Fields Winter 2021

Recall that, if you take an infinite number of terms, the series for $\sin z$ and the function itself $f(z)=\sin z$ are equivalent representations of the same thing for all real numbers $z$, (in fact, for all complex numbers $z$). This is not always true. More commonly, a series is only a valid, equivalent representation of a function for some more restricted values of $z$. The technical name for this idea is convergence--the series only "converges" to the value of the function on some restricted domain.

Find the power series for the function $f(z)=\frac{1}{1+z^2}$. Then, using the Mathematica worksheet from class (vfpowerapprox.nb) as a model, or some other computer algebra system like Sage or Maple, explore the convergence of this series. Where does your series for this new function converge? Can you tell anything about the region of convergence from the graphs of the various approximations? Print out a plot and write a brief description (a sentence or two) of the region of convergence.

Note: As a matter of professional ettiquette (or in some cases, as a legal copyright requirement), if you use or modify a computer program written by someone else, you should always acknowledge that fact briefly in whatever you write up. Say something like: “This calculation was based on a (name of software package) program titled (title) originally written by (author) copyright (copyright date).

The following are 2 different represtations for the $\textbf{same}$ state on a quantum ring $$\left|{\Phi}\right\rangle = \sqrt\frac{1}{2}\left|{2}\right\rangle -\sqrt\frac{ 1}{4}\left|{0}\right\rangle +i\sqrt\frac{ 1}{4}\left|{-2}\right\rangle$$ $$\Phi(\phi) \doteq \sqrt {\frac{1}{8 \pi r_0}} \left( \sqrt{2}e^{i 2 \phi} -1 + ie^{-i 2 \phi} \right)$$

1. Write down the matrix representation for the same state.

2. With all 3 representations, calculate the probability that a measurement of $L_z$ will yield $0\hbar$, $-2\hbar$, $2\hbar$.

3. If you measured the $z$-component of angular momentum to be $2\hbar$, what would the state of the particle be immediately after the measurement is made?

4. What is the probability that a measurement of energy, $E$, will yield $0\frac{\hbar^2}{I}$?,$2\frac{\hbar^2}{I}$?,$4\frac{\hbar^2}{I}$?

5. If you measured the energy of the state to be $2\frac{\hbar^2}{I}$, what would the state of the particle be immediately after the measurement is made?

Author Information
Dustin Treece and Corinne Manogue
Learning Outcomes