Ring Function

• group Working with Representations on the Ring

group Small Group Activity

30 min.

Working with Representations on the Ring
Central Forces 2023 (3 years)
• assignment Lines in Polar Coordinates

assignment Homework

Lines in Polar Coordinates
Central Forces 2023 (3 years)

The general equation for a straight line in polar coordinates is given by: $$r(\phi)=\frac{r_0}{\cos(\phi-\delta)}$$ where $r_0$ and $\delta$ are constant parameters. 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. $y=-3x+2$

• group Wavefunctions on a Quantum Ring

group Small Group Activity

30 min.

Wavefunctions on a Quantum Ring
Central Forces 2023 (2 years)
• 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.
• assignment Visualization of Wave Functions on a Ring

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.
• 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 Potential vs. Potential Energy

assignment Homework

Potential vs. Potential Energy
Static Fields 2023 (6 years)

In this course, two of the primary examples we will be using are the potential due to gravity and the potential due to an electric charge. Both of these forces vary like $\frac{1}{r}$, so they will have many, many similarities. Most of the calculations we do for the one case will be true for the other. But there are some extremely important differences:

1. Find the value of the electrostatic potential energy of a system consisting of a hydrogen nucleus and an electron separated by the Bohr radius. Find the value of the gravitational potential energy of the same two particles at the same radius. Use the same system of units in both cases. Compare and the contrast the two answers.
2. Find the value of the electrostatic potential due to the nucleus of a hydrogen atom at the Bohr radius. Find the gravitational potential due to the nucleus at the same radius. Use the same system of units in both cases. Compare and contrast the two answers.
3. Briefly discuss at least one other fundamental difference between electromagnetic and gravitational systems. Hint: Why are we bound to the earth gravitationally, but not electromagnetically?

• face Quantum Reference Sheet

face Lecture

5 min.

Quantum Reference Sheet
Central Forces 2023 (6 years)
• assignment Working with Representations on the Ring

assignment Homework

Working with Representations on the Ring
Central Forces 2023 (3 years)

The following are 3 different representations for the $\textbf{same}$ state on a quantum ring for $r_0=1$ $$\left|{\Phi_a}\right\rangle = i\sqrt{\frac{ 2}{12}}\left|{3}\right\rangle - \sqrt{\frac{ 1}{12}}\left|{1}\right\rangle +\sqrt{\frac{ 3}{12}}e^{i\frac{\pi}{4}}\left|{0}\right\rangle -i\sqrt{\frac{ 2}{ 12}}\left|{-1}\right\rangle +\sqrt{\frac{ 4}{12}}\left|{-3}\right\rangle$$ $$\left| \Phi_b\right\rangle \doteq \left( \begin{matrix} \vdots \\ i\sqrt{\frac{ 2}{12}}\\ 0 \\ -\sqrt{\frac{ 1}{12}} \\ \sqrt{\frac{ 3}{12}}e^{i\frac{\pi}{4}} \\ -i\sqrt{\frac{ 2}{12}}\\ 0 \\ \sqrt{\frac{4}{12} }\\ \vdots \end{matrix}\right) \begin{matrix} \leftarrow m=0 \end{matrix}$$ $$\Phi_c(\phi) \doteq \sqrt{\frac{1}{24 \pi}} \left( i\sqrt{2}e^{i 3 \phi} -e^{i\phi} +\sqrt{3}e^{i\frac{\pi}{4}} -i \sqrt{2} e^{-i\phi} + \sqrt{4}e^{-i 3 \phi} \right)$$

1. With each representation of the state given above, explicitly calculate the probability that $L_z=-1\hbar$. Then, calculate all other non-zero probabilities for values of $L_z$ with a method/representation of your choice.
2. Explain how you could be sure you calculated all of the non-zero probabilities.
3. If you measured the $z$-component of angular momentum to be $3\hbar$, what would the state of the particle be immediately after the measurement is made?
4. With each representation of the state given above, explicitly calculate the probability that $E=\frac{9}{2}\frac{\hbar^2}{I}$. Then, calculate all other non-zero probabilities for values of $E$ with a method of your choice.
5. If you measured the energy of the state to be $\frac{9}{2}\frac{\hbar^2}{I}$, what would the state of the particle be immediately after the measurement is made?

• assignment Helix

assignment Homework

Helix

Integration Sequence

Static Fields 2023 (6 years)

A helix with 17 turns has height $H$ and radius $R$. Charge is distributed on the helix so that the charge density increases like (i.e. proportional to) the square of the distance up the helix. At the bottom of the helix the linear charge density is $0~\frac{\textrm{C}}{\textrm{m}}$. At the top of the helix, the linear charge density is $13~\frac{\textrm{C}}{\textrm{m}}$. What is the total charge on the helix?

• Central Forces 2023 (3 years) 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?