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
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 (Chapters 31 and
32)
Thursday,
January 28: Supersymmetric quantum mechanics (Chapter 33)
Tuesday,
February 2: The Pauli equation and the Dirac operator (Chapter 34)
Thursday,
February 4: Lagrangian methods and the path integral (Chapter 35)
Tuesday,
February 9 :
Thursday,
February 11:
Tuesday,
February 16:
Thursday,
February 18:
Tuesday,
February 23:
Thursday,
February 25:
Tuesday,
March 9:
Thursday,
March 11:
Tuesday,
March 16:
Thursday,
March 18:
Tuesday,
March 23:
Thursday,
March 25:
Tuesday,
March 30:
Thursday,
April 1:
Tuesday,
April 6 :
Thursday,
April 8:
Tuesday,
April 13:
Thursday,
April 15:
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