assignment Homework

Spin-1 Eigenvectors
eigenvectors 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.

group Small Group Activity

30 min.

Right Angles on Spacetime Diagrams
Theoretical Mechanics (4 years)

Special Relativity

Students take the inner product of vectors that lie on the spacetime axis to show that they are orthogonal. To do the inner product, students much use the Minkowski metric.

group Small Group Activity

30 min.

Hydrogen Probabilities in Matrix Notation
Central Forces 2023 (2 years)

group Small Group Activity

60 min.

Expectation Value and Uncertainty for the Difference of Dice
Quantum Fundamentals 2023 (3 years)

group Small Group Activity

10 min.

Matrix Representation of Angular Momentum
Central Forces 2023 (2 years)

group Small Group Activity

60 min.

Going from Spin States to Wavefunctions
Quantum Fundamentals 2023 (2 years)

Wavefunctions quantum states probability amplitude histograms matrix notation of quantum states Arms representation

Arms Sequence for Complex Numbers and Quantum States

Completeness Relations

Students review using the Arms representation to represent states for discrete quantum systems and connecting the Arms representation to histogram and matrix representation. The student then extend the Arms representation to begin exploring the continuous position basis.

group Small Group Activity

30 min.

Outer Product of a Vector on Itself
Quantum Fundamentals 2023 (2 years)

Projection Operators Outer Products Matrices

Completeness Relations

Students compute the outer product of a vector on itself to product a projection operator. Students discover that projection operators are idempotent (square to themselves) and that a complete set of outer products of an orthonormal basis is the identity (a completeness relation).

keyboard 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.