## Activity: Wavefunctions on a Quantum Ring

Central Forces 2023 (2 years)

Consider the following normalized quantum state on a unit ring ($r_0 = 1$): $$\Phi(\phi)=\sqrt\frac{8}{3 \pi } \sin^{2}\left( 3\,\phi \right)\cos \left( \phi \right)$$

1. Translate this state into eigenfunction, bra/ket, and matrix representations. Remember that you can use any of these representations in the following calculations.

2. What is the probability that the particle can be found in the region $0<\phi< \frac{\pi}{4}$? In the region $\frac{\pi}{4}<\phi< \frac{3 \pi}{4}$?
3. What is the expectation value of $L_z$ in this state?
4. The wave function and it's probability density are plotted below. What features of these graphs (if any) tell you the expectation value of $L_z$ in this state?

• 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?

• group Working with Representations on the Ring

group Small Group Activity

30 min.

##### Working with Representations on the Ring
Central Forces 2023 (3 years)
• 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 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?
• assignment_ind Normalization of the Gaussian for Wavefunctions

assignment_ind Small White Board Question

5 min.

##### Normalization of the Gaussian for Wavefunctions
Periodic Systems 2022

Fourier Transforms and Wave Packets

Students find a wavefunction that corresponds to a Gaussian probability density.
• group Fourier Transform of a Plane Wave

group Small Group Activity

5 min.

##### Fourier Transform of a Plane Wave
Periodic Systems 2022

Fourier Transforms and Wave Packets

• group Fourier Transform of a Derivative

group Small Group Activity

10 min.

##### Fourier Transform of a Derivative
Periodic Systems 2022

Fourier Transforms and Wave Packets

• assignment Fourier Transform of Cosine and Sine

assignment Homework

##### Fourier Transform of Cosine and Sine
Periodic Systems 2022
1. Find the Fourier transforms of $f(x)=\cos kx$ and $g(x)=\sin kx$.
2. Find the Fourier transform of $g(x)$ using the formula for the Fourier transform of a derivative and your result for the Fourier transform of $f(x)$. Compare with your previous answer.
3. In quantum mechanics, the Fourier transform is the set of coefficients in the expansion of a quantum state in terms of plane waves, i.e. the function $\tilde{f}(k)$ is a continuous histogram of how much each functions $e^{ikx}$ contributes to the quantum state. What does the Fourier transform of the function $\cos kx$ tell you about which plane waves make up this quantum state? Write a sentence or two about how this makes sense.
• face Systems of Particles Lecture Notes

face Lecture

10 min.

##### Systems of Particles Lecture Notes
Central Forces 2023 (3 years)
• assignment Charge on a Spiral

assignment Homework

##### Charge on a Spiral
Static Fields 2023 (2 years) A charged spiral in the $x,y$-plane has 6 turns from the origin out to a maximum radius $R$ , with $\phi$ increasing proportionally to the distance from the center of the spiral. Charge is distributed on the spiral so that the charge density increases linearly as the radial distance from the center increases. At the center of the spiral the linear charge density is $0~\frac{\textrm{C}}{\textrm{m}}$. At the end of the spiral, the linear charge density is $13~\frac{\textrm{C}}{\textrm{m}}$. What is the total charge on the spiral?

Learning Outcomes