## Activity: Matrix Representation of Angular Momentum

Central Forces 2023 (2 years)

The operator $\hat{L}_z$ that represents the $z$-component of angular momentum, the operator $\hat{L}^2$ that represents the total angular momentum, and the operator $\hat{H}$ that represents the energy for the rigid rotor (a particle confined to the unit sphere) have eigenvalues given by \begin{align} \hat{L}_z \left|{\ell, m}\right\rangle &=m\hbar \left|{\ell, m}\right\rangle \\ \hat{L}^2 \left|{\ell, m}\right\rangle &=\ell(\ell+1)\hbar^2 \left|{\ell, m}\right\rangle \\ \hat{H} \left|{\ell, m}\right\rangle &=\frac{\hbar^2}{2I}\, \ell(\ell+1)\left|{\ell, m}\right\rangle \end{align} Find the matrix representations for these operators.

• keyboard Kinetic energy

keyboard Computational Activity

120 min.

##### Kinetic energy
Computational Physics Lab II 2022

Students implement a finite-difference approximation for the kinetic energy operator as a matrix, and then use numpy to solve for eigenvalues and eigenstates, which they visualize.
• group Working with Representations on the Ring

group Small Group Activity

30 min.

##### Working with Representations on the Ring
Central Forces 2023 (3 years)
• keyboard Position operator

keyboard Computational Activity

120 min.

##### Position operator
Computational Physics Lab II 2022

Students find matrix elements of the position operator $\hat x$ in a sinusoidal basis. This allows them to express this operator as a matrix, which they can then numerically diagonalize and visualize the eigenfunctions.
• 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.
• group Hydrogen Probabilities in Matrix Notation

group Small Group Activity

30 min.

##### Hydrogen Probabilities in Matrix Notation
Central Forces 2023 (2 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?

• group Wavefunctions on a Quantum Ring

group Small Group Activity

30 min.

##### Wavefunctions on a Quantum Ring
Central Forces 2023 (2 years)
• face Systems of Particles Lecture Notes

face Lecture

10 min.

##### Systems of Particles Lecture Notes
Central Forces 2023 (3 years)
• face Equipartition theorem

face Lecture

30 min.

##### Equipartition theorem
Contemporary Challenges 2022 (4 years)

This lecture introduces the equipartition theorem.
• face Ideal Gas

face Lecture

120 min.

##### Ideal Gas
Thermal and Statistical Physics 2020

These notes from week 6 of Thermal and Statistical Physics cover the ideal gas from a grand canonical standpoint starting with the solutions to a particle in a three-dimensional box. They include a number of small group activities.

Learning Outcomes