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
  • SWBQ: Review Angular Momentum
  • SWBQ: Spinning Top Precession
  • Bicycle wheel precession demo (background for Spin Precession)
  • 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

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
  • 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
  • Finding orthogonal vectors
  • Normalizing quantum state vectors
Hour 16: Bases
  • Using different bases to express quantum states
  • $| \pm\rangle_n$ in spherical coordinates

SGA: Finding Unknown States

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

Measurement

Hour 23: Statistics
  • Average and Standard Deviation
  • Expectation Value and Uncertainty
  • Commutation
  • Uncertainty Relations

Time Dependence

  • For Time Independent Hamiltonians
  • SGA: Conditions for Time Dependent Probabilities
Hour 28: Precession
  • Spin Precession
Hour 29: UNNAMED
  • Rabi Oscillations

Intro to Spatial Potentials

  • Wavefunctions
  • Translating between Dirac Notation and Wavefunction language
  • SGA: Operators and Functions
  • 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
  • SGA: Representations of ISW Superposition States

Review

Hour 38: Review

Optional Topics

Can be skipped

  • Neutrino Oscillation (SGA) 40 min FIXME
  • Magnetic Resonance
  • EPR Paradox
  • Schrodinger's Cat

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