Spin-1 Eigenvectors

eigenvectors

Finding if $S_{x}, \; S_{y}, \; and \; S_{z}$ Commute

Small Group Activity

30 min.

Finding if $S_{x}, \; S_{y}, \; and \; S_{z}$ Commute
Quantum Fundamentals 2023 (3 years)
Kinetic energy

Computational Activity

120 min.

Kinetic energy
Computational Physics Lab II 2022
finite 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.
Frequency

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.
Operators & Functions
Small Group Activity

30 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.
Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)

Kinesthetic

30 min.

Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)
Quantum Fundamentals 2021
arms complex numbers phase rotation reflection math

Arms Sequence for Complex Numbers and Quantum States
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.
Position operator

Computational Activity

120 min.

Position operator
Computational Physics Lab II 2022
quantum 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.
Changing Spin Bases with a Completeness Relation

Small Group Activity

10 min.

Changing Spin Bases with a Completeness Relation
Quantum Fundamentals 2023 (3 years)
Completeness Relations Quantum States

Completeness Relations
Students work in small groups to use completeness relations to change the basis of quantum states.
Spin 1/2 with Arms

Kinesthetic

10 min.

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

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

Small Group Activity

30 min.

Quantum Expectation Values
Quantum Fundamentals 2023 (3 years)

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