Quantum Mechanics involves conceptually difficult material—a particle's position is usually indeterminate before measurement—and the notions of superposition and of entanglement are not easy to grasp. But these ideas do not require high level mathematics to express them. Quantum Mechanics for Everyone (Physics 008x) presents the conceptual ideas of Quantum Mechanics (primarily superposition, indeterminacy, and entanglement) in a way that's accessable to everyone. The course uses detailed animations and descriptive videos throughout. It starts by covering the classical mechanics of magnets leading to an understanding of how one can separate small magnetic current loops by shooting them through an inhomogeneous magnetic field. Then we move into the quantum world by examining what happens in real Stern-Gerlach experiments. This requires a foray into probability and it brings home the idea of indeterminacy; it follows many ideas from the text of Daniel Styer. Wheeler's delayed choice experiments, the EPR experiment, an introduction to quantum seeing in the dark, and Bell's inequality are next taught in the context of these Stern-Gerlach experiments. Next up is the quantum mechanics of light. Here, a quantum model of light is developed and used as a framework to describe the Mach-Zender interferometer, quantum seeing in the dark, and the Hong-Ou-Mandel experiment; it follows closely a text by Richard Feynman. All this is done using only high-school-level math. We have summarized more ideas about the course in a paper published in The Physics Teacher.

Why is it important to bring quantum mechanics to the general population? Because quantum mechanics is one of the highest intellectual achievements of mankind. It is a pity that such a small percentage of people understand details of how it works. By offereing rigorous classes at a level that a wide population of students can follow the logic, we empower many more people to understand the bizarre features of the quantum world. As we enter the second quantum revolution, this becomes increasingly important to carry out. This course development was partially supported by the National Science Foundation.

This class will be rerun starting on June 28, 2021. Just click the above link to register. You can also see course evaluations.

List of Videos and Tutorials for Quantum Mechanics for Everyone

Course Trailer
Module 1: Introduction to the quantum world

Module 1, Lecture 1: Let's get small
Module 1, Lecture 2: The fallacy of physics phobia
Module 1, Lecture 3: The Quantum world around us
Module 1, Lecture 4: Pushes and pulls, twists and turns
Module 1, Lecture 5: Opposites attract
Module 1, Lecture 6: Fat Arrows are Fields
Module 1, Lecture 7: Twisting magnetic needles
Module 1, Lecture 8: Projections to calculate forces
Module 1, Lecture 9: Right is Correct
Module 1, Lecture 10: Current loop moves a bar magnet
Module 1, Lecture 11: Current loop is an effective magnet
Module 1, Lecture 12: To and Fro
Module 1, Lecture 13: The complicated motion we call precession
Module 1, Lecture 14: Motion of a current loop
Module 1, Lecture 15: Current loop in a magnetic field
Module 1, Lecture 16: The classical Stern-Gerlach experiment
Module 1, Simulation 1: Classical Stern-Gerlach experiment
Module 1, Simulation 2: The Stern-Gerlach experiment with atoms
Module 1, Simulation 3: The Stern-Gerlach analyzer
Module 1, Simulation 4: The repeated measurements with the Stern-Gerlach analyzer
Module 1, Simulation 5: Perpendicular Stern-Gerlach analyzers
Module 1, Simulation 6: Three perpendicular Stern-Gerlach analyzers
Module 1, Simulation 7: Analyzers at an angle
Module 1, Lecture 17: Fate or Chance?
Module 1, Simulation 8: Three coins
Module 1, Simulation 9: Birthday simulator
Module 1, Lecture 18: Penney's game
Module 1, Simulation 10: Play to win!
Module 1, Lecture 19: The rotating Stern-Gerlach analyzer

Advanced quantum mechanics with spins

Module 2, Simulation 1: Introduction to the Stern-Gerlach analyzer loop
Module 2, Simulation 2: Experiments with the Stern-Gerlach analyzer loop
Module 2, Simulation 3: The analog of the two-slit experiment with the Stern-Gerlach analyzer loop
Module 2, Lecture 1: Two analyzer loops
Module 2, Simulation 4: Watching at the gates
Module 2, Simulation 5: Choosing the experiment after the loop
Module 2, Lecture 2: Can we trick nature?
Module 2, Lecture 3: Einstein's spooky action at a distance
Module 2, Simulation 6: The Einstein-Podolsky-Rosen Paradox
Module 2, Lecture 4: Proving Einstein Wrong
Module 2, Simulation 7: The Bell Theorem 1
Module 2, Simulation 8: The Bell Theorem 2
Module 2, Simulation 9: The Bell Theorem 3
Module 2, Simulation 10: The Bell Theorem 4
Module 2, Lecture 5: Nuclear Magnetic Resonance
Module 2, Lecture 6: Hey Doc, the MRI uses quantum mechanics!

The quantum mechanics of light

Module 3, Lecture 1: Photons are Particles Not Waves
Module 3, Simulation 1: Simulation of Single Photon Detection
Module 3, Lecture 2: Rules, Rules, Those Quantum Rules
Module 3, Lecture 3: Rules, Rules, Those Quantum Rules II
Module 3, Lecture 4: Quantum Simulations to Help You Understand
Module 3, Simulation 2: Practicing with the quantum rules for photons
Module 3, Simulation 3: Detecting light inside glass
Module 3, Simulation 4: Light reflecting off of glass
Module 3, Lecture 5: Reflections, Reflections, Everywhere
Module 3, Lecture 6: From Here to Infinity
Module 3, Simulation 5: Using the compute engine to study light traveling from a source to a detector
Module 3, Simulation 6: Using the compute engine to examine how the pattern changes with a varying slit width
Module 3, Lecture 7: Squeezing Photons Through a Narrow Slit
Module 3, Simulation 7: How does light of different colors travel through slits of the same thickness?
Module 3, Lecture 8: Your Turn to Explore the Quantum Rules
Module 3, Simulation 8: Free exploration computer tutorial
Module 3, Lecture 9: Double Trouble
Module 3, Lecture 10: Photons can do that?
Module 3, Simulation 9: Now explore the two-slit experiment using our quantum rules
Module 3, Simulation 10: Next, examine what an experiment would look like
Module 3, Lecture 11: It changes when I watch
Module 3, Simulation 11: What happens when we watch
Module 3, Lecture 12: Imperfect detectors
Module 3, Lecture 13: Three times the charm
Module 3, Simulation 12: Open exploration of the three slit experiment
Module 3, Lecture 14: Photon self-reflection
Module 3, Lecture 15: Through the Looking Glass
Module 3, Simulation 13: Demonstration of the Lens

Advanced quantum ideas with light

Module 4, Lecture 1: Looking Without Seeing
Module 4, Lecture 2: Do you see what I see?
Module 4, Simulation 1: Two-slit interaction free measurement
Module 4, Lecture 3: Anything you can do I can do better
Module 4, Lecture 4: Entangling at a Distance, Part I
Module 4, Simulation 2: Quantum theory for the Mach-Zehnder interferometer
Module 4, Lecture 5: Entangling at a Distance, Part 2
Module 4, Simulation 3: Comparing lengths with the Mach-Zehnder interferometer
Module 4, Lecture 6: Half Full or Half Empty?
Module 4, Simulation 4: Quantum Seeing in the Dark with an interferometer
Module 4, Lecture 7: Catch me if you can
Module 4, Lecture 8: The quantum eraser
Module 4, Simulation 5: Introduction to Polarization of Light
Module 4, Simulation 6: Rotating the Polarization of Light
Module 4, Lecture 9: Stern-Gerlach-like experiments with photons
Module 4, Simulation 7: Repeated measurements with polarizers
Module 4, Simulation 8: Free exploration of Polarization of Light
Module 4, Lecture 10: Show Me the Light
Module 4, Simulation 9: Rotating the polarization
Module 4, Lecture 11: Polarized Mach-Zehnder interferometer
Module 4, Simulation 10: Polarizing Beam Splitter
Module 4, Lecture 12: That is just so weird
Module 4, Simulation 11: Quantum seeing in the dark I
Module 4, Simulation 12: Quantum seeing in the dark II
Module 4, Lecture 13: Two of the same are different
Module 4, Simulation 13: Compound probabilities and partial reflection
Module 4, Simulation 14: Hong-Ou-Mandel
Module 4, Lecture 14: The wide world of quantum correlations

There is a problem in teaching quantum mechanics to undergraduates. Many students feel that the course is more of a mathematics course than a physics course and they feel lost in their conceptual understanding of what the material really means. Furthermore, most quantum courses are taught in the coordinate representation, which goes against the trend we teach students in earlier courses about vectors being defined independent of the choice of the coordinate system. To try to resolve these disconnects, we have designed a new quantum mechanics course that addresses these issues.

The course focuses first on the conceptual ideas of superposition, indeterminancy and entanglement. This is done using the materials from Quantum Mechanics for Everyone, but supplements them with instruction on Dirac notation and spin operators. Next, we develop four fundamental operator identities: (i) the Leibniz, or product rule, for commutators; (ii) the Hadamard lemma (and the braiding relation and exponential re-ordering identity); (iii) the Baker-Campbell-Hausdorff formula; and (iv) the exponential disentangling identity. While the first three are fairly well known identities, the fourth one is used primarily in quantum optics. Armed with these identities, students are empowered to solve many of the problems that appear in conventional classes, but in a way that helps develop "operator sense" (indeed this way of working with quantum mechanics should be thought of as "operator mechanics"). We use the Schrödinger factorization method to solve eigenvalue problems, which allows us to cover many more such problems than those covered in conventional quantum mechanics classes; indeed, the Schrödinger equation in coordinate space is only introduced in the 12th week of the class. This allows us to focus heavily on different experimental results (see the word cloud to the right, which highlights the different experiments). The course ends with a discussion of second quantization of photons, leading up to a description of how LIGO works. This allows students to both have a clear functional understanding of how photons are different from just dim light, as well as have them understand how something as sophisticated as gravitational wave detection can be carried out.

Quantum Mechanics is being offered through edX starting August 15, 2021 (and annually afterwards). It provides a strong foundation for quantum sensing applications and can serve as the basis for a concentrated training in quantum sensing for the quantum-enabled workforce. The course development was partially supported by the National Science Foundation. We have plans on testing this use with engineering students at pilot test sites (North Carolina State University is considering doing this in Spring of 2022 and we have had some discussions with Stanford University as well).

The videos of the course are available in a youtube channel; just be cautioned that the course has significantly more material than what is in the videos only. The syllabus is also available. Note that the first four weeks of videos are from the Quantum Mechanics for Everyone couse. This list starts with Module 5 of the undergraduate course.

List of Videos for the Undergraduate Course

Introductory video for the course
Videos for Modules 1 to 4 are in Quantum Mechanics for Everyone (under the "Everyone" tab above)
Module 5: Quantum Operators and their identities

Module 5, Lecture 1: Spin Matrix Representations
Module 5, Lecture 2: Spin Matrix Identities
Module 5, Lecture 3: Canonical Commutation Relation (Part I)
Module 5, Lecture 3: Canonical Commutation Relation (Part II)
Module 5, Lecture 4: Hadamard Lemma

Module 6: Quantum Mechanics of the Simple Harmonic Oscillator

Module 6, Lecture 1: Factorization Method for a Free particle on a Circle
Module 6, Lecture 2: Heisenberg Uncertainty Relation
Module 6, Lecture 3: Schrödinger Factorization method for the Simple Harmonic Oscillator
Module 6, Lecture 4: The Baker-Campbell-Hausdorff Identity
Module 6, Lecture 5: Coherent States
Module 6, Lecture 6: The Simple Harmonic Oscillator in Three Dimensions

Module 7: Quantum Angular Momentum and Hydrogen

Module 7: Lecture 1: Commutation Relations and Angular Momentum (Part I)
Module 7: Lecture 1: Commutation Relations and Angular Momentum (Part II)
Module 7: Lecture 2: Spherical Harmonics via Operators
Module 7: Lecture 3: Schrödinger Factorization Method I
Module 7: Lecture 4: Schrödinger Factorization Method II
Module 7: Lecture 5: The Two-Body Problem
Module 7: Lecture 6: Radial Momentum
Module 7: Lecture 7: Isotropic Simple Harmonic Oscillator
Module 7: Lecture 8: Hydrogen Energy Eigenstates
Module 7: Lecture 9: Coulomb Wavefunctions
Module 7: Lecture 10: Cartesian Factorization of Hydrogen

Module 8: Quantum Approximation Methods

Module 8: Lecture 1: First Order Perturbation Theory
Module 8: Lecture 2: Second Order Perturbation Theory for the Energy
Module 8: Lecture 3: Particle in a Box and the Schrödinger Equation

Module 9: Quantum Time Evolution

Module 9: Lecture 1: Time Evolution and the Trotter Product Formula
Module 9: Lecture 2: Cyclotron Resonance

Module 10: Photons and LIGO

Module 10: Lecture 1: Quantization of Light and Photons
Module 10: Lecture 2: Verifying Single Photons Exist
Module 10: Lecture 3: Mach-Zehnder Interferometer and Single-Photon Interference

One challenge with quantum mechanics instruction is that single-particle quantum mechanics and many-body problems often use quite different methods in their solution. When working with the differential equation form of the Schrödinger equation, the standard approach is to use a series-based solution (following a sequence of changes of variables and of wavefunction ansatzes) called the Fröbenius method. This method is rarely used in modern research problems. On the other hand, much of many body physics and quantum field theory employs methodologies from complex analysis. In our work, we are showing how one can combine training in complex analysis and its use in quantum mechanics earlier in the curriculum. We feel this is most applicable at the graduate level, where students are sophisticated enough to work with complex variables.

One might be concerned that trying to do this will require an inordinate amount of complex analysis instruction prior to being able to use the Laplace method. But that is not the case. One can essentially prove Cauchy's theorem from Stoke's theorem, with the condition that the curl vanishes being identical to the Cauchy-Riemann equations. The residue theorem follows almost immeiately then by calculating the integrals of powers with contours that wind around the origin once. The only technical detail remaining is to describe the logarithm, nonintegal powers, single valuedness, and branch cuts. This is straightforward to do by adopting standard complex analysis rules for how to determine the phase relative to a reference point in the complex plane. This ends up being all of the material needed for the complex analysis background. The methodology is completed by carefully working out the asymptotic analysis of integrals over contours using stationary phase and other related methods. Note that these are techniques that are widely used in modern research, so it is useful to take the time to teach students about this.

Unfortunately, we have not been able to incorporate this into instruction yet. But, courtesy of the pandemic, we do have a series of videos recorded for a second semester graduate level physics course. This course is a bit different from a standard graduate physics course as we discuss the Schrödinger factorization method and also discuss more many body topics including second quantization for light and electrons. We include the full video list here, but encourage those interested in the course to consult the course webpage which has much more details.

The videos of the course are available in a youtube channel; just be cautioned that the course has significantly more material than what is in the videos only.

List of Videos for the Graduate Course

Introductory video for the course
Lecture 1: Review of spin operators, Pauli matrices, and Pauli matrix identities
Lecture 2: The five quantum operator identities: (i) Leibnitz, (ii) Hadamard, (iii) Baker-Campbell-Hausdorff, (iv) exponential reordering (braiding) and (v) exponential disentangling
Lecture 3: Algebraic derivation of the simple harmonic oscillator wavefunction
Lecture 4: Coherent states
Lecture 5: Squeezed states
Lecture 6: Schroedinger factorization method
Lecture 7: Rotations and angular momentum
Lecture 8: Spherical harmonics, the algebraic way
Lecture 9: Center of mass, two-body problem, and translation operator in spherical harmonics
Lecture 10: Hydrogen via the factorization method in coordinate space
Lecture 11: Cartesian factorization method for hydrogen and the momentum-space wavefunctions
Lecture 12: Addition of angular momenta I
Lecture 13: Addition of angular momenta II
Lecture 14: Nondegenerate perturbation theory
Lecture 15: Wigner-Brillouin perturbation theory
Lecture 16: Degenerate Perturbation Theory I: Formalism development
Lecture 17: Degenerate Perturbation Theory II: Summary and Atomic Fine Structure
Lecture 18: Degenerate Perturbation Theory III: Hydrogen atom in an external magnetic field
Lecture 19: Degenerate Perturbation Theory IV: The Stark effect and spin examples
Lecture 20: Introduction to scattering I
Lecture 21: Introduction to scattering II
Lecture 22: 3d scattering and the generalized optical theorem
Lecture 23: Partial wave scattering
Lecture 24: Collisional (Feshbach) resonances
Lecture 25: The time-dependent Schroedinger equation (Part I)
Lecture 25: The time-dependent Schroedinger equation (Part II)
Lecture 26: The interaction representation (Part I)
Lecture 26: The interaction representation (Part II)
Lecture 27: Cyclotron Resonance
Lecture 28: An exact Time-Ordered Product (Part I)
Lecture 28: An exact Time-Ordered Product (Part II)
Lecture 29: Time-dependent perturbation theory
Lecture 30: Landau-Zener diabatic passage
Lecture 31: Simulating quantum problems with ion traps (Part I)
Lecture 31: Simulating quantum problems with ion traps (Part II)
Lecture 32: Fermi's golden rule and the sudden approximation
Lecture 33: Photoproduction of Hydrogen
Lecture 34: What is a photon?
Lecture 35: How LIGO works
Lecture 36: Fermionic creation and annihilation operators
Lecture 37: Applications of creation/annihilation operators
Lecture 38: The Hubbard Model
Lecture 39: Two-site Hubbard model solution
Lecture 40: Nagaoka Ferromagnetism
Lecture 41: Antiferromagnetism

At Georgetown University, we offer a mathematical physics class to sophomores in their second semester. Loosely speaking, the learning goal of this class is to transform students from being technicians of math to practitioners of math. In other words, provide a deeper, richer understanding of the math they will use, along with illustrating how math is used within physics. The course has a number of applications and demonstrations interweaved into the material. It covers the topics of calculus, the integral theorems of multivariable calculus, complex numbers and complex analysis through the residue theorems, linear algebra, differential equations and Fourier series. I felt it important to emphasize the way math is used in physics by also incorporating a laboratory each week. The class is offered on edx annually starting in January. It was first offered in 2021. Mathematical and Computational Methods (Physics 155/155x).

At Georgetown, the class has been offered as a flipped class since 2018. The main benefits of the flipped class format is that students are able to get more practice with solving problems and that they get immediate feedback as to whether they are right or wrong on a particular problem. Both are likely to help with learning. The course also provides optional opportunities for using retrieval practice to help long-term retention of the material. The math covered here is all that is needed for an operator-based quantum mechanics course. (It does not teach the Fröbenius method, so it does not completely prepare for traditional quantum mechanics classes). The development of this course was partially supported by the National Aeronautical and Space Administration. You can also see course evaluations.

The videos of the course are available in a youtube channel; just be cautioned that the course has significantly more material than what is in the videos only. The syllabus is also available.

List of Videos for the Math Methods Course

Lecture 0: Devious Math Tricks
Module 1: Calculus Review

Lecture 1: Irrational Numbers Exist (Part I)
Lecture 1: Ratios (Part II)
Lecture 2: Zeno's Paradox and the Geometric Series (Part I)
Lecture 2: Archimedes, Pi, and Exhaustion (Part II)
Lecture 2: The Taylor Series (Part III)
Problem of the Week 1: The Railroad Rail Problem (Setup)
Problem of the Week 1: The Railroad Rail Problem (Solution)
Lecture 3: The Sum of Powers of Integers (Part I)
Lecture 3: How to define the Integral (Part II)
Lecture 3: Numerical Quadrature (Part III)
Problem of the Week 2: The Infinite Resistor Network Problem
Lecture 4: Tangents and Derivatives (Part I)
Lecture 4: The Greeks and Sine Tables (Part II)
Lecture 4: John Napier and Logarithm Tables (Part III)
Lecture 5: The Fundmental Theorem of Calculus (Part I)
Lecture 5: Integration Strategies (Part II)
Lecture 5: Inverse Functions (Part III)
Lecture 6: Integration of Polynomials and Rational Functions (Part I)
Lecture 6: Gaussian and Frullani's Integrals (Part II)
Lecture 7: Arc length (Part I)
Lecture 7: Surfaces and Volumes (Part II)
Lecture 8: The Vanishing Sphere
Problem of the Week 3: A Frullani Integral
Lecture 9: Feynman Integration: Gaussians and Powers (Part I)
Lecture 9: Feynman Integration: Other Examples (Part II)
Problem of the Week 4: Rolling Races

Module 2: Vector Calculus

Lecture 10: Vector-Valued Functions
Lecture 11: Surface Integrals
Lecture 12: The Divergence Theorem
Problem of the Week 5: More on Surface Integrals
Lecture 13: The Line Integral and the Curl
Lecture 14: Stokes Theorem
Lecture 15: Line Integral and the Gradient (Part I)
Lecture 15: The Laplacian (Part II)
Problem of the Week 6: Coaxial Cable
Lecture 16: Example of Laplace's Equation (Part I)
Lecture 16: The Relaxation method (Part II)
Lecture 17: The Laplacian in Cylindrical Coordinates (Part I)
Lecture 17: The Laplacian in Spherical Coordinates (Part II)
Problem of the Week 7: The Divergence Theorem

Module 3: Complex Variables

Lecture 18: Complex Numbers (Part I)
Lecture 18: Cauchy-Riemann Equations (Part II)
Problem of the Week 8: Product of Sines
Lecture 19: Cauchy's Theorem
Lecture 20: The Residue Theorem
Problem of the Week 9 (Bonus): The Calculus of Residues

Module 4: Linear Algebra

Lecture 21: Row Reduction
Problem of the Week 9: Partial Fractions
Lecture 22: How to Calculate Determinants
Lecture 23: Matrix Inverse
Lecture 24: Vector Spaces
Lecture 25: Gram-Schmidt Orthogonalization
Problem of the Week 10: Functions as Vectors
Lecture 26: Change of Basis
Lecture 27: Eigenvalues and Eigenvctors
Lecture 28: Physical Examples of Eigenvalues and Eigenvctors
Problem of the Week 11: Coupled Oscillations

Module 5: Differential Equations

Lecture 29: First-order Linear Differential Equations
Lecture 30: First-order Nonlinear Differential Equations
Problem of the Week 12: Making a Total Derivative
Lecture 31: Physics Examples of First-order Differential Equations
Lecture 32: Arbitrary-Order Linear Differential Equations
Lecture 33: Arbitrary-Order Linear Differential Equations with Constant Coefficients
Lecture 34: Method of Undetermined Coefficients
Lecture 35: Introduction to Differential Geometry and the Frenet-Serret Apparatus

Module 6: Fourier Series

Lecture 36: The Dirichlet Problem on Heat Flow
Lecture 37: Separation of Variables
Lecture 38: Applications of Poisson's Theorem