- eigenvectors
*group*Finding if $S_{x}, \; S_{y}, \; and \; S_{z}$ Commute*group*Small Group Activity30 min.

##### Finding if \(S_{x}, \; S_{y}, \; and \; S_{z}\) Commute

Quantum Fundamentals 2023 (3 years)*keyboard*Kinetic energy*keyboard*Computational Activity120 min.

##### Kinetic energy

Computational Physics Lab II 2022finite difference hamiltonian quantum mechanics particle in a box eigenfunctions

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.*assignment*Frequency*assignment*Homework##### Frequency

Quantum Mechanics Time Evolution Spin Precession Expectation Value Bohr Frequency Quantum Fundamentals 2023 (3 years) Consider a two-state quantum system with a Hamiltonian \begin{equation} \hat{H}\doteq \begin{pmatrix} E_1&0\\ 0&E_2 \end{pmatrix} \end{equation} Another physical observable \(M\) is described by the operator \begin{equation} \hat{M}\doteq \begin{pmatrix} 0&c\\ c&0 \end{pmatrix} \end{equation} 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.*group*Operators & Functions*group*Small Group Activity30 min.

##### Operators & Functions

Quantum Fundamentals 2023 (3 years) Students are asked to:- Test to see if one of the given functions is an eigenfunction of the given operator
- See if they can write the functions that are found not to be eigenfunctions as a linear combination of eigenfunctions.

*accessibility_new*Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)*accessibility_new*Kinesthetic30 min.

##### Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)

Quantum Fundamentals 2021 Students, working in pairs, represent two component complex vectors with their left arms. Through a short series of instructor led prompts, students move their left arms to show how various linear transformations affect each complex component.*keyboard*Position operator*keyboard*Computational Activity120 min.

##### Position operator

Computational Physics Lab II 2022quantum mechanics operator matrix element particle in a box eigenfunction

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*Changing Spin Bases with a Completeness Relation*group*Small Group Activity10 min.

##### Changing Spin Bases with a Completeness Relation

Quantum Fundamentals 2023 (3 years) Students work in small groups to use completeness relations to change the basis of quantum states.*accessibility_new*Spin 1/2 with Arms*accessibility_new*Kinesthetic10 min.

##### Spin 1/2 with Arms

Quantum Fundamentals 2023 (2 years)Quantum State Vectors Complex Numbers Spin 1/2 Arms Representation

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.*assignment*Spin Three Halves Time Dependence*assignment*Homework##### Spin Three Halves Time Dependence

Quantum Fundamentals 2023 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, \(\hat{H} = \alpha \hat{S}_x \doteq \frac{\alpha\hbar}{2}\begin{pmatrix} 0 & \sqrt{3} & 0 & 0\\ \sqrt{3} & 0 & 2 & 0\\ 0 & 2 & 0 & \sqrt{3} \\ 0 & 0 & \sqrt{3} & 0 \end{pmatrix}\)- Find the energy eigenvalues and energy eigenstates for the system.
- Find \(|\psi(t)\rangle\).
- List the outcomes of all possible measurements of \(S_x\) and find their probabilities. Explicitly identify any probabilities that depend on time.
- List the outcomes of all possible measurements of \(S_z\) and find their probabilities. Explicitly identify any probabilities that depend on time.

*group*Quantum Expectation Values-
Quantum Fundamentals 2023
The operator \(\hat{S}_x\) for spin-1 may be written as:
defined by:
\[\hat{S}_x=\frac{\hbar}{\sqrt{2}}
\begin{pmatrix}
0&1&0\\ 1&0&1 \\ 0&1&0 \\
\end{pmatrix}
\]
- Find the eigenvalues and eigenvectors of this matrix. Write the eigenvectors as both matrices and kets.
- Confirm that the eigenstates you found give probabilities that match your expectation from the Spins simulation for spin-1 particles.