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### 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)

- Introduction to Complex Numbers (Lecture, 20 minutes)
- Visualizing Complex Numbers (Kinesthetic Activity)

Write these lecture notes, emphasize Euler's formula

#### Matrix Manipulations (45 minutes)

- Review of matrix manipulations (Lecture, 45 minutes)

#### Introduction to Bra-ket notation (15 minutes)

Ian is turning this into an activity

- Introduction to bra-ket notation (Lecture, ?? minutes) - Emphasize normalization and finding components. Add a good SWBQ or small group activity.

#### Inner Products and Norms (10 minutes)

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

Write these lecture notes. Emphasize complex vectors.

#### Linear Transformations (1 hr)

- Linear Transformations Activity (Small Group Activity)

#### Tangible Metaphor for Complex Vectors (10 minutes)

- Visualizing Complex Two Component Vectors (Kinesthetic Activity)

#### Matrix Components (40 min)

- Finding Matrix Elements (Small Group Activity) Work in comments about Fourier series into wrap-up.

#### Rotation Matrices in 2 and 3 Dimensions (10 min)

- Rotation Matrices in 2 and 3 Dimensions (Lecture, ?? minutes).

#### Eigenvalues & Eigenvectors (2 hr)

- Eigenvectors and Eigenvalues Activity (Small Group Activity)

**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)

- Properties of linear vector spaces (Optional Lecture, ?? minutes)

#### Special Properties of Hermitian Matrices (40 min)

- Properties of Hermitian matrices (Lecture, 10 minutes without proof-30 minutes with proofs)
- Diagonalization of matrices (Lecture, 15 minutes)
- Diagonalizing Matrices (Small Group Activity).

#### Commuting Matrices (10 min)

- Properties of commuting matrices (Lecture, ?? minutes)

### 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. 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)

- The Magnetic Dipole Moment (Lecture, 10 minutes)

- Lorentz Force and Work on a Rectangular Loop (Small groups, 55 minutes We are actively working to shorten and streamline this activity)

- The Lorentz Force Law (SWBQ, 5 minutes)

##### Circular Loop

- The Magnetic Dipole Moment of a Ring (Lecture, 10 minutes)
- The Angular Momentum of a Ring
- The Lorentz Force Law
- The Energy of the Spinning Ring

#### Classical Probabilities (50 minutes)

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.

- Introduction to Probability (Lecture, 20 minutes)

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

- The Birthday Problem (Activity, 5 minutes)
- Dice Rolling Lab (Activity, 25 minutes)
- How Choosing the # Trials vs. # Experiments Changes Results (Lecture, 15 minutes)
- How Choosing the # Trials vs. # Experiments Changes Results 2 (Lecture, 15 minutes)

#### The Stern-Gerlach Experiment (30 minutes)

- Postulates of Quantum Mechanics (Lecture, 10 minutes)
- The Stern-Gerlach Experiment (Lecture, 10 minutes)
- Expected vs. Observed Results of the S.G. Experiment (Lecture, 10 minutes)

### Unit: Quantum States

#### Quantum State Vectors, Probability (45 minutes)

- Introduction to the SPINS program (Lecture, 10 minutes)
- Terminology for the Stern-Gerlach Experiment (Lecture, 5 minutes)
- Probabilities in the z-direction for a Spin-$\frac{1}{2}$ System (Simulation, 10 minutes)
- Probabilities for Different Spin-$\frac{1}{2}$ Stern Gerlach Analyzers (Simulation, 15 minutes)
- Probabilities for Different Spin-1 Stern Gerlach Analyzers (Simulation, 15 minutes)
- Stern-Gerlach Experiment Probabilities (SWBQ, 5 minutes)
- Postulate Four of Quantum Mechanics (Lecture, 10 minutes)
- Guessing the Form of Spin along x in Terms of z (SWBQ, 5 minutes)
- Dimensionality of the Ket Vector Space (Lecture, 10 minutes)
- Deriving spin-up with x Orientation in z-ket Basis (Lecture, 25 minutes)
- Computing $_{x}\langle -\vert+ \rangle_{x}$ (SWBQ, 5 minutes)

#### Review

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

- Bra-ket notation (Lecture, 5 minutes)
- "Taking the Square of the Norm" (SWBQ, 5 minutes)
- Finding the Square of the Norm of a Complex Number (SWBQ, 5 minutes)
- Representing Complex Values in Polar & Rectangular Form (SWBQ, 5 minutes)
- Relative Phases in Quantum States (Lecture, 10 minutes)
- Normalization of Quantum State Vectors (Lecture, 5 minutes)

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

- Quantum Friend (Small groups, 30 minutes)
- Finding Unknown States Leaving the Oven in a Spin-$\frac{1}{2}$ System (Simulation, 30 minutes)
- Determining how a $\vec{B}$ Field Changes a Spin-$\frac{1}{2}$ Particle's State (Simulation, 30 minutes)

Move this to before Rabi oscillations

- Finding Unknown States Leaving the Oven in a Spin-1 System (Simulation, 20 minutes)

### Unit: Quantum Operators

#### Introduction to the Projection Postulate (60 minutes)

- How Making Measurements Affects Results in Quantum Systems (Lecture, 10 minutes)
- The Projection Postulate of Quantum Mechanics (Lecture, 5 minutes)
- The Projection Operator & Wave Function Collapse (Lecture, 30 minutes)

#### Interferometers (1 hour--some can be homework)

- The Outer Product (Lecture, 10 minutes)
- Computing Several Outer Products (SWBQ, 5 minutes)
- Results of a State Collapse (SWBQ, 10 minutes)
- The Completeness Relation (lecture, 10 minutes)
- Writing the Completeness Relation for the 3-state Case (SWBQs, 5 minutes)
- Analyzing a Spin-$\frac{1}{2}$ Interferometer (Theoretical) (Simulation, 45 minutes)
- Analyzing a Spin-$1$ Interferometer (Simulation, 30 minutes)

#### Hermitian Operators (1 hour)

- The $S_{z}$ Operator (SWBQ, 5 minutes)
- Naming Quantum Operators (Class Activity, 5 minutes)
- Computing the $S^{2}$ Operator for the Spin-$\frac{1}{2}$ System (SWBQ, 5 minutes)
- The $S^{2}$ Operator (lecture, 15 minutes)
- The Hamiltonian (lecture, 10 minutes)

#### Commutators & Uncertainty Relations (1 hour 10 minutes)

- Commutators & Commuting Operators (lecture, 15 minutes)
- Finding if $S_{x},\: S_{y},\; and \; S_{z}$ commute (activity, 5 minutes)
- Expectation Values & Quantum Uncertainty (lecture, 15 minutes)
- Practice Finding Expectation Values (small groups, 20 minutes)

#### Density Operator (Optional - Advanced, 50 minutes)

- Introducing the Density Operator (lecture, 20 minutes)
- Practice With Outer Product Matrix Properties (SWBQ, 5 minutes)
- Describing the Sample Oven in the Stern-Gerlach Experiment Mathematically (Small groups, 15 minutes)
- Applications for the Density Function (Lecture, 10 minutes)

### Unit: Time Dependence

#### Time Evolution (2 hours 30 minutes)

- The Schrodinger Equation (lecture, 25 minutes)
- Factoring out Overall Phases (SWBQ, 5 minutes)
- Analyzing the Probabilities of Time-evolved States (small groups, 110 minutes)

#### Rabi Oscillations & Magnetic Resonance (1 hour 40 minutes)

- Ehrenfest's Theorem (lecture, 10 minutes)
- Spin Precession (lecture, 5 minutes)
- The Correspondence Principle (lecture, 5 minutes)
- Rabi Flopping (lecture, 40 minutes)
- Magnetic Resonance (lecture, 40 minutes)