Unit: Complex Numbers and Linear Algebra

Note: This Unit does not appear in McIntyre's textbook. However, it is essential that students have a basic familiarity with complex numbers, including Euler's formula, and with linear algebra, including finding eigenvalues and eigenvectors. This Unit represents our approach to that content (7 class hours-10 class hours if you include all of the optional content.)

Complex Numbers (20 minutes)

FIXME Write these lecture notes, emphasize Euler's formula

Matrix Manipulations (45 minutes)

Introduction to Bra-ket notation (15 minutes)

FIXME Ian is turning this into an activity

Inner Products and Norms (10 minutes)

  • Calculating Inner Products and Normalization of Vectors (Lecture, 10 minutes)

FIXME Write these lecture notes. Emphasize complex vectors.

Linear Transformations (1 hr)

Tangible Metaphor for Complex Vectors (10 minutes)

Matrix Components (40 min)

Rotation Matrices in 2 and 3 Dimensions (10 min)

Eigenvalues & Eigenvectors (2 hr)

If time allows, cover any or all of the following content. Otherwise, summarize the results and/or work the content into the rest of the Spins course.

Properties of Linear Vector Spaces (30 min)

Special Properties of Hermitian Matrices (40 min)

Commuting Matrices (10 min)

Unit: Background and Stern-Gerlach Experiment

Classical Spin

Two approaches are available to introduce the classical spin ideas. One approach, circular loop, is intended for students who had experienced with integral of abstract vectors. The other approach, rectangular loop, is intended for students who only had experienced with explicit vector components. If the time is in essence, quote the result from either approach. FIXME Move rectangular loop to homework, write course notes for students and include here for Circular loop from Day 2,3 2016. Comment that adopters who want solutions to any homework can contact us by email. Also solutions to spins hw1, problem 4 e, f.

Rectangular Loop(1 hr or a bit more)
Circular Loop

Classical Probabilities (50 minutes)

FIXME This topic was originally part of Spins Lab 1. There are two problems with that lab. First, that students don't understand the mean as a weighted average. Second, that the effect of binning data is not clear. We ignored these problems for 15 years and you can safely do so also.

FIXME We are working on the following activities intended to address the problems above.

The Stern-Gerlach Experiment (30 minutes)

Unit: Quantum States

Quantum State Vectors, Probability (45 minutes)

Review

You may need to sprinkle these review topics several times each throughout the course!

Unknown Quantum States (2 hours--some can be homework)

Unit: Quantum Operators

Introduction to the Projection Postulate (60 minutes)

Interferometers (1 hour--some can be homework)

Hermitian Operators (1 hour)

Commutators & Uncertainty Relations (1 hour 10 minutes)

Density Operator (Optional - Advanced, 50 minutes)

Unit: Time Dependence

Time Evolution (2 hours 30 minutes)

Rabi Oscillations & Magnetic Resonance (1 hour 40 minutes)


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