## Quantum Measurement and Spin

This course is centered around the quantum mechanical two state system. This course is the first of the quantum paradigms and introduces students to quantum mechanics by leading off with the postulates of quantum mechanics. A central theme of this course is having students perform some simulated experiments with Stern-Gerlach devices and interpret their results. Students learn about quantum measurement, Dirac Notation, vector spaces, quantum mechanical operators, commuting observables, incompatible observables, the uncertainty principle, and time-evolution of quantum states. Spin 1 systems are also introduced as a second context for exploring and interpreting Stern-Gerlach experiments. The course ends with introductions to special topics like Rabi oscillations, neutrino oscillations, magnetic resonance, the EPR paradox, and quantum computing. (more...)

**Textbook:** Quantum Mechanics: A Paradigms Approach—-a textbook that follows the paradigms approach. The chapters that are relevant to the Quantum Measurement and Spins course are: the appendix on linear algebra: Linear Algebra and Matrices, Chapter 1: Stern-Gerlach Experiments, Chapter 2: Operators and Measurement, Chapter 3: Schrödinger Time Evolution and Chapter 4: Quantum Spookiness, and the Instructor's Guide

**Sample Syllabus:** 425_syllabus_wi10.doc

## Course Contents

### Unit: Stern-Gerlach Experiments

#### Historic Stern-Gerlach Experiment (70 minutes)

- Introduction to the course (Lecture, 15 minutes)
- Quantum Postulates (Small Whiteboard Activity, 5 minutes)
- Discussion of Historic Stern-Gerlach Experiment (Lecture, 30 minutes)
- Components of the SG Experiment (Small Whiteboard Activity, 5 minutes)
- Discussion of Intrinsic Spin (Lecture, 15 minutes)

#### Probability and Statistics (30 minutes)

- Introduction to statistics (Lecture, 10 minutes)
- Dice (Small Group Activity, 20 minutes)

#### Using Stern-Gerlach Observations to Build Quantum State Formalism (245 minutes)

- Introduction to Stern Gerlach simulation (Lecture, 10 minutes)
- SPINS Lab 1 (Small Group Activity, 60 minutes)
- Outcome from Successive Stern-Gerlach Devices (Lecture, 40 minutes)
- Guessing forms for spin along x in terms of z (Small Whiteboard Activity, 5 minutes)
- State Formalism (Small Group Activity, 15 minutes)
- Building Quantum State Formalism (Lecture, 40 minutes)
- Calculating 'crossed polarizers' effect (Small Group Activity, 15 minutes)

- SPINS Lab 2 (Small Group Activity, 60 minutes)

#### Generalized Spin Systems (20 minutes)

- Spin N systems (Lecture, 20 minutes)

### Unit: Operators and Measurement

#### Operators, Eigenvalues, and Eigenvectors

- Operators (Lecture, 10 minutes)
- Matrix Representation of Spin Operators (Small Group Activity, 45 minutes)
- Spin projections in a general direction (Lecture, 10 minutes)
- Projection Operators (Small Group Activity, XX minutes)
- Here is a resource for the labs and the operator activity - a pdf file of a talk regarding these specific sequences: nwaps_oct2010.pdf

#### Expectation Value, Standard Deviation, Commutation, and Uncertainty

- Expectation Value and Standard Deviation (Lecture, 20 minutes)
- Commutation and Uncertainty (Lecture, 20 minutes)

#### Spin-1 Systems

- Spin 1 systems (Lecture, 10 minutes)
- SPINS Lab 3 (Small Group Activity, 100 minutes)

#### The $S$ and $S^2$ Vectors (XX minutes)

- The $S$ and $S^2$ Vectors (Lecture, 20 minutes)

### Unit: Schrödinger Time Evolution

#### Schrodinger equation, energy eigenvalues and eigenstates

- Schrodinger equation (Lecture, 60 minutes)
- Time Evolution Introduction (Small Group Activity, 20 minutes)
- Visualizing Complex Time Dependence for Spin 1/2 Systems (Kinesthetic Activity, XX minutes)
- Energy eigenvalues and eigenstates (Lecture, 60 minutes)
- Quantum Time Evolution (Small Group Activity, 20 minutes)

#### Spin Precession

- Spin Precession (Lecture, 80 minutes)
- Time-dependent spin vector (Small Group Activity, 40 minutes)
- SPINS Lab 4 (Small Group Activity, 100 minutes)

#### Neutrino Oscillation

*Optional topic - can be skipped*

- Neutrino Oscillation (Lecture, 30 minutes)
- Neutrino Oscillations (Small Group Activity, 20 minutes)

#### Magnetic Resonance

*Optional topic - can be skipped*

- Here are slides that can be used for this topic (Lecture, 40 minutes): nmr.pdf
- Appropriate homework problems can be chosen from the end of Chapter 3, none explicitly address this topic

### Unit: Quantum Spookiness

*Optional topics - can be skipped*

(What we have done from 2008-2010 is have a guest lecturer speak on one of the following topics)

#### Quantum Clocks

- Here is a scan of the lecture notes from the invited guest lecturer (Lecture, 60 minutes): 425_guest_lecture.pdf
- There have not been homework problems designed for this guest lecture

#### EPR Paradox

- Here are slides addressing this topic (Lecture, 60 minutes): bells_inequality.pdf
- Additional reading of interest: a paper giving an Sherlock Holmes-type analogy to Bell's theorem bells_theorem.pdf
- Appropriate homework problems are at the end of Chapter 4

#### Schrodinger Cat Paradox

- There are no current lecture notes for this topic, addressed in chapter 4 (Lecture, 60 minutes)
- Here are slides for the related topic of Quantum Cryptography: quantum_cryptography.pdf
- Appropriate homework problems are at the end of Chapter 4

### Activities Included

- All activities for Spins