### Course Overview

: Course Site Under Construction.

The “Quantum Measurements and Spin” course is built upon the quantum mechanical two state system. The first of the Quantum Paradigms, this course introduces students to quantum mechanics by beginning with the postulates of quantum mechanics and how the postulates are used to gather information about quantum mechanical systems. The common spin-up and spin-down state vectors with x,y, and z-orientation will be derived, and the general state vector $\vert \psi (\theta,\phi ) \rangle$ will also be introduced. Throughout the class, students perform several simulated experiments with virtual Stern-Gerlach devices and interpret their results (Spins OSP software). Operators that correspond to physical observables in quantum experimentation are then presented; students will learn in particular about spin operators, projection operators, the density operator, and the Hamiltonian. Important physical relations among these quantum operators will also be made using the commutators, uncertainty relations, and expectation values. Spin 1 systems are also introduced as an additional context for exploring and interpreting Stern-Gerlach experiments. The time evolution of quantum states using the Schrodinger Equation will also be explored to investigate time dependence in probabilities, uncertainties, and expectation values. The course ends with introductions to special topics of spin precession, Rabi oscillations, and magnetic resonance. (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

### Activities Included

### Unit: Introduction

#### 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 are currently working on activities that address these problems. In the meantime, we ignored them for 15 years and you can safely do so also.

- The Birthday Problem (Activity, 5 minutes)
- Dice Rolling Lab (Activity, 25 minutes)
- Introduction to Probability (Lecture, 20 minutes)
- How Choosing the # Trials vs. # Experiments Changes Results (Lecture, 15 minutes)
- How Choosing the # Trials vs. # Experiments Changes Results 2 (Lecture, 15 minutes)

#### Classical Spin (1 hr or a bit more)

- The Lorentz Force Law (SWBQ, 5 minutes)
- Lorentz Force and Work on a Rectangular Loop (Small groups, 55 minutes We are actively working to shorten and streamline this activity)
- The Magnetic Dipole Moment (Lecture, 10 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)

#### Finding Expressions for Unknown Quantum States (2 hours 15 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)
- Finding Unknown States Leaving the Oven in a Spin-1 System (Simulation, 20 minutes)
- Quantum Friend (Small groups, 30 minutes)

### Unit: Quantum Operators

#### Projection (60 minutes)

- Analyzing a Spin-$\frac{1}{2}$ Interferometer (Experimental) (Simulation, 15 minutes)
- How Making Measurements Affects Results in Quantum Systems (Lecture, 10 minutes)
- Postulate 5 of Quantum Mechanics (Lecture, 5 minutes)
- The Projection Operator & Wave Function Collapse (Lecture, 30 minutes)

#### Measurement (2 hours 25 minutes)

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

#### 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)

### Unit: Quantum Spookiness

*Optional topics - can be skipped* (In 2008-2010 we had a gues lecturer speak on one of the following topics. Their lecture notes or powerpoint slides are included as a resource.)

#### Quantum Clocks

- Here is a scan of the lecture notes from the invited guest lecturer (Lecture, 60 minutes): 425_guest_lecture.pdf

#### 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)
- Appropriate homework problems are at the end of Chapter 4

#### Quantum Cryptography

- Here are slides for the related topic of Quantum Cryptography: quantum_cryptography.pdf

### Spins SWBQs

- In process of editing. See Master SWBQ List at “Pedagogy” → “Small Whiteboard Questions”