Introduction to Quantum Mechanics: Mathematics GU4392 (spring 2021)

Tuesday and Thursday 4:10-5:25pm

Class lectures will initially be online-only as in the fall, although depending on circumstances I may try and arrange for a classroom and move the class to a "hybrid" model with students who are on campus able to attend in-person.  Zoom links are available on Courseworks.  I'll likely continue to also have videos of the class available at this Youtube channel.

I'll have online office hours after each class starting around 5:30, there are separate Zoom links for the office hours on Courseworks.  I will be in the office most of the time Tuesday and Thursday. If you are in the area and able to come in for an in-person office hour, please contact me (or just try and stop by Math 421 Tuesday or Thursday).

This course will be a continuation of the fall course (Math GU4391) which is usually a prerequisite.  If you did not take the fall course, but have a good background in quantum mechanics, you should also be able to take the course, but consult with me first.  This course is open to both undergraduate and graduate students.  It can be taken independently and in addition to any of the Physics department courses on quantum mechanics.

Lecture Notes/Book

The lecture notes from previous versions of this course have been turned into a book, see here.   During this course I expect to be revising some of the material in the book, and maybe adding some new chapters.  The most recent version will always be available here

During the spring semester I expect to cover roughly the material in chapter 27-47 of the book.  The overall goal of the class is to explain at least the fundamental ingredients that go into the Standard Model.  We will however only explain how to calculate things for free quantum fields, avoiding the very large and complicated topic of how to do calculations for interacting quantum fields.

Before each class, please try and read the chapter in the syllabus announced for that class and come prepared with questions about whatever you don't understand.  I hope to devote much of the time in each class to going over material students are finding confusing, rather than repeating everything that is in the notes.

Problem Sets and Exams

There will be problem sets due roughly every week or two, a midterm and a final project.  Use of notes is allowed during the midterm exam.  Grading will be based on these according to: 50 % final project, 25 % midterm exam, 25 % problem sets.

First problem set: due Tuesday, January 26.
Problems 1 and 2 in appendix B.12, problems 1 and 2 in appendix B.13 and problem 1 in appendix B.14 of the book

Second problem set: due Tuesday, February 9.
Problems 3 and 4 in appendix B.14 and problems 1 and 2 in appendix B.15 of the book

Third problem set: due Tuesday, February 23.
Problems 1 and 2 in appendix B.16 and problems 1 and 2 in appendix B.17 of the book

Fourth problem set/midterm: due Tuesday, March 9 (due at class time, work on these for no more than three hours)
Problems 3 and 4 in appendix  B.17 of the book

Fifth problem set: due Tuesday, March 16.
Problems 1,2,3 and 4 in appendix B.18 of the book

Sixth problem set: due Tuesday, March 30
Problems 1 and 2 in appendix B.19 of the book

Seventh problem set: due Thursday, April 15
Problems 1,2,3 in appendix B.20 and Problems 1,2,3 in appendix B.21 of the book

Tentative Schedule of Lectures

Chapter numbers correspond to the course textbook, Quantum Theory, Groups and Representations.

Tuesday, January 12: Overview, review of fall semester, summary of Chapters 24-26.

Thursday, January 14: Fermionic oscillators, Weyl and Clifford algebras (Chapters 27 and 28)

Tuesday, January 19: Clifford algebras and geometry (Chapter 29)

Thursday, January 21: Anti-commuting variables and pseudo-classical mechanics (Chapter 30)

Tuesday, January 26: Fermionic quantization and spinors (Chapter 31)

Thursday,  January 28: Fermionic quantization and spinors (Chapters 32)

Tuesday, February 2: Supersymmetric quantum mechanics (Chapter 33)

Thursday, February 4: The Pauli equation and the Dirac operator (Chapter 34)

Tuesday,  February 9 :  Lagrangian methods and the path integral (Chapter 35)

Thursday, February 11:  Multi-particle systems (Chapter 36)

Tuesday, February 16:  Field quantization (Chapter 37)

Thursday, February 18:  Review of symmetries and quadratic operators (Chapters 24-26)

Tuesday, February 23:  Symmetries and non-relativistic quantum fields (Chapter 38)

Thursday, February 25:  Minkowski space and the Lorentz group (Chapter 40)

Tuesday, March 9:  Representations of the Lorentz group (Chapter 41)

Thursday, March 11:  The Poincaré group and its representations (Chapter 42)

Tuesday, March 16: The Klein-Gordon equation and scalar quantum fields (Chapter 43)

Thursday, March 18: Symmetries and propagators for relativistic scalar quantum fields (Chapters 43 and  44)
Tuesday, March 23: Symmetries and propagators for relativistic scalar quantum fields (Chapters 43 and  44)

Thursday, March 25: U(1) Gauge symmetry and electromagnetic fields (Chapter 45)

Tuesday, March 30:  Quantization of the electromagnetic field: the photon (Chapter 46) part I

Thursday, April 1: Quantization of the electromagnetic field: the photon (Chapter 46) part II

Tuesday, April 6 : The Dirac equation and spin 1/2 fields (Chapter 47) part I

Thursday, April 8 : The Dirac equation and spin 1/2 fields (Chapter 47) part II

Tuesday, April 13:  An introduction to the Standard model (Chapter 48)

Thursday, April 15:  Student presentations

Other Textbooks

The following gives suggestions for other reading material most relevant to the fall course.  I will try and later add here some other suggestions for the material of we'll be covering in the spring.

A standard physics textbook at the upper-undergraduate to beginning graduate level should be available to consult for more details about the physics and some of the calculations we will be studying.  A good choice for this is

Principles of Quantum Mechanics, by Ramamurti Shankar. Springer, 1994.

which does a good job of carefully working out the details of many calculations. Two good undergraduate-level texts are

A Modern Approach to Quantum Mechanics, John S. Townsend, University Science Books, 2000.
Introduction to Quantum Mechanics, David J. Griffiths, Prentice-Hall, 1995.

Several suggestions for standard physics textbooks that provide good references for some of the topics we will be considering are:

Quantum Mechanics, Volume 1, by Cohen-Tannoudji, Diu and Laloe. Wiley, 1978
The Feynman Lectures on Physics, Volume III, by Richard Feynman. Addison-Wesley 1965.
Lectures on Quantum Mechanics, Gordon Baym.
Quantum Mechanics, Volumes 1 and 2, Albert Messiah.
Quantum Mechanics, Volume 1, Kurt Gottfried.
Introduction to Quantum Mechanics, David J. Griffiths.
Quantum Mechanics and the Particles of Nature: an Outline for Mathematicians, Sudbery. Cambridge 1986 (unfortunately out of print)

Some other books at various levels that students might find helpful:

More mathematical:

An Introduction to Quantum Theory, by Keith Hannabuss. Oxford, 1997.
Quantum Mechanics for Mathematicians, by Leon Takhtajan. AMS, 2008.
Lectures on Quantum Mechanics for Mathematics Students, by L.D. Fadeev and O.A. Yakubovskii. AMS, 2009.
Linearity, Symmetry and Prediction in the Hydrogen Atom, Stephanie Singer, Springer, 2005. (On Springerlink at this URL)

Some more from the physics side, available via Springerlink:

Quantum Mechanics, Franz Schwabl.
Lectures on Quantum Mechanics, Jean-Louis Basdevant.
Quantum Mechanics, Daniel Bes.

A classic:

The Theory of Groups and Quantum Mechanics, Hermann Weyl.

Also emphasizing groups and representations, but covering mostly different material:

Group theory and physics, Shlomo Sternberg.

More advanced, from the point of view of analysis:

Mathematical Methods in Quantum Mechanics, Gerald Teschl

Recommended sources on Lie groups, Lie algebras and representation theoy:

Naive Lie Theory, John Stillwell
Groups and Symmetries: From Finite Groups to Lie Groups, Yvette Kossmann-Schwarzbach
An Elementary Introduction to Groups and Representations, Brian C. Hall
Lie groups, Lie algebras and representations, Brian C. Hall
Representation Theory, Constantin Teleman

Online Resources

Lecture notes for a course on Quantum Computation, John Preskill (especially Chapters 1-3)

Previous Courses

Introduction to Quantum Mechanics, Fall 2012: Math W4391
Introduction to Quantum Mechanics, Spring 2013: Math W4392
Introduction to Quantum Mechanics, Fall 2014: Math W4391
Introduction to Quantum Mechanics, Spring 2015: Math W4392
Introduction to Quantum Mechanics, Fall 2020: Math GU4391