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
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.
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
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
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.
January 12: Overview, review of fall semester, summary of Chapters
January 14: Fermionic oscillators, Weyl and Clifford algebras
(Chapters 27 and 28)
January 19: Clifford algebras and geometry (Chapter 29)
January 21: Anti-commuting variables and pseudo-classical
mechanics (Chapter 30)
January 26: Fermionic quantization and spinors (Chapter 31)
January 28: Fermionic quantization and spinors (Chapters 32)
February 2: Supersymmetric quantum mechanics (Chapter 33)
February 4: The Pauli equation and the Dirac operator (Chapter 34)
February 9 : Lagrangian methods and the path integral
February 11: Multi-particle systems (Chapter 36)
February 16: Field quantization (Chapter 37)
February 18: Review of symmetries and quadratic operators
February 23: Symmetries and non-relativistic quantum fields
February 25: Minkowski space and the Lorentz group (Chapter
March 9: Representations of the Lorentz group (Chapter 41)
March 11: The Poincaré group and its representations
March 16: The Klein-Gordon equation and scalar quantum fields
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)
March 25: U(1) Gauge symmetry and electromagnetic fields (Chapter
March 30: Quantization of the electromagnetic field: the
photon (Chapter 46) part I
April 1: Quantization of the electromagnetic field: the photon
(Chapter 46) part II
April 6 : The Dirac equation and spin 1/2 fields (Chapter 47)
April 8 : The Dirac equation and spin 1/2 fields (Chapter 47) part
April 13: An introduction to the Standard model (Chapter 48)
April 15: Student presentations
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
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,
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,
Quantum Mechanics, Volumes 1 and 2,
Quantum Mechanics, Volume 1,
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:
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.
Linearity, Symmetry and Prediction
in the Hydrogen Atom, Stephanie Singer, Springer, 2005. (On
Springerlink at this
Some more from the physics side, available via Springerlink:
Mechanics, Franz Schwabl.
on Quantum Mechanics, Jean-Louis Basdevant.
Mechanics, Daniel Bes.
The Theory of Groups and Quantum
Mechanics, Hermann Weyl.
Also emphasizing groups and representations, but covering mostly
Group theory and physics,
More advanced, from the point of view of analysis:
Methods in Quantum Mechanics, Gerald Teschl
Recommended sources on Lie groups, Lie algebras and representation
Lie Theory, John Stillwell
and Symmetries: From Finite Groups to Lie Groups,
Introduction to Groups and Representations, Brian C. Hall
Lie groups, Lie algebras and
representations, Brian C. Hall
notes for a course on Quantum Computation, John Preskill
(especially Chapters 1-3)
to Quantum Mechanics, Fall 2012: Math W4391
to Quantum Mechanics, Spring 2013: Math W4392
to Quantum Mechanics, Fall 2014: Math W4391
to Quantum Mechanics, Spring 2015: Math W4392
to Quantum Mechanics, Fall 2020: Math GU4391