## Summary Learning Goals

**Goal 1:** Predict and contrast the results of Stern Gerlach experiments for classical and quantum particles

**Goal 2:** Use linear algebra concepts (inner products, change of basis, eigenvectors, etc.) to describe quantum systems

**Goal 3:** Calculate probabilities, expectations values, and uncertainties for various experiments

**Goal 4:** Describe the effects of a measurement on the state vector for a quantum system

**Goal 5:** Make predictions about the time evolution of quantum states and probabilities

**Goal 6:** Appreciate quantum ”spookiness”(??) (Distinguish purely quantum behavior from classical)

**Goal 7:** Use spin-1/2 systems as a productive analogy for generic quantum systems and the infinite square well

**Goal 8:** Use and explain the connections between Dirac, matrix, and wavefunction notations to perform calculations

## Course Schedule of Topics

### Classical Review: Spin and Magnetic Moment

**Hour 1: Administrative**

**Hour 2: Classical Angular Momentum**- SWBQ: Review Angular Momentum
- SWBQ: Spinning Top Precession
- Bicycle wheel precession demo (background for Spin Precession)

**Hour 3: Magnetic Moment**- Force & Torque on a magnetic moment
- SWQB Sequence: Spinning Charged Sphere in a Magnetic Field
- Intro to Stern-Gerlach Experiment

### Math Bits: Linear Algebra and Complex Numbers

**Hour 5 Complex Numbers**

**Hour 6 Linear Transformations**

**Hour 7 Bra Ket and Matrix Elements**

**Hours 8-9 Eigenvectors and Eigenvalues**

**Hour 10 Special Matricies**

### Spin Systems

- WCD: What is a model?
- WCD: What is a state?
- Introduce the State Postulate
- SGA: Probabilities of Stern-Gerlach Measurement Simulation

- Introduce quantum state vector
- Probabilities as norm squares of coefficients
- The Probability Postulate
- Representing quantum states with arms

**Hour 13: UNNAMED**- Introduce Spin 1 and General Quantum States

**Hour 14: Determining Spins State from Data**- Finding coefficients from SG data, in general
- Finding $| \pm\rangle_x$ and $| \pm\rangle_y$ from SG data
- Real space vs. Hilbert space and visualizing quantum states with graphs and arms

**Hour 15: Orthogonal and Normal Vectors**- Finding orthogonal vectors
- Normalizing quantum state vectors

**Hour 16: Bases**- Using different bases to express quantum states
- $| \pm\rangle_n$ in spherical coordinates

**Hour 17: Determining States from Data**

SGA: Finding Unknown States

### Operators

**Hour 18: Collapse & Projection Operators**- Projection Operators
- The Projection Postulate

**Hour 19 UNNAMED**

**Hour 20: Observables**- Properties of Hermitian Matrices: Orthogonality, Real Eigenvectors
- Spin Eigenvalue Equations
- SGA: Matrix Form of Spin Operators

**Hour 21: Total Spin**- Calculating matrix elements of operators
- Spin Vector $\vec{S}$
- $\hat{S}^2$ operator

**Hour 22: (Optional) EPR Paradox**

### Measurement

**Hour 23: Statistics**- Average and Standard Deviation
- Expectation Value and Uncertainty

**Hour 24: Communtation**- Commutation

**Hour 25: Uncertainty Relations**- Uncertainty Relations

### Time Dependence

**Hour 26: Solving Schrodinger**- For Time Independent Hamiltonians

**Hour 27: Stationary States**- SGA: Conditions for Time Dependent Probabilities

**Hour 28: Precession**- Spin Precession

**Hour 29: UNNAMED**- Rabi Oscillations

### Intro to Spatial Potentials

**Hour 30: Classical Probability Density**

**Hour 31: Wavefunction Representation**- Wavefunctions
- Translating between Dirac Notation and Wavefunction language

- SGA: Operators and Functions

**Hours 33: Infinite Square Well**- Separate Variables
- Solve Spatial Diff. Eq.
- Apply Boundary Conditions & Normalization

- Interpret Eigenstates
- Covariation with parameters
- Visualize with PhET

- Solve Time Dependence
- Visualize with PhET
- Review Stationary States

**Hours 36-37: Representations of ISW**- SGA: Representations of ISW Superposition States

### Review

**Hour 38: Review**

### Optional Topics

*Can be skipped*

- Neutrino Oscillation (SGA)
*40 min* - Magnetic Resonance
- EPR Paradox
- Schrodinger's Cat