Introduction to Quantum Mechanics:
Mathematics GU4391 (fall 2020)
Tuesday and Thursday 4:10-5:25pm
Class lectures will be online-only for now (NOT in Math 203
as listed by the registrar). Zoom links are available on
Courseworks. As an experiment, I'll also have videos of the
class available at this
Teaching Assistant: Davis Lazowski
Davis will have office/help room hours Monday 5-6pm and Thursday
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 an introduction to the subject of quantum
mechanics, from a perspective emphasizing the role of Lie groups and
their representations. Most of the standard material and
examples from conventional physics courses will be covered, but with
much greater attention to the mathematical ideas behind the standard
formalism and usual calculational techniques. There will
be a continuation of this course (Math GU4392) in the spring
covering more advanced material.
No specific background in physics will be assumed, although an
elementary physics course of some kind would be helpful. The
mathematical prerequisites are multi-variable calculus (as in
Calculus IV), and Linear Algebra. 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 fall semester I expect to cover roughly the material
in the first 23 chapters of the book. 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, a midterm and a
final exam. The final exam is scheduled for Tuesday, December 22,
4:10-7pm. Use of notes is allowed during the exams.
Grading will be based on these according to: 50 % final exam, 25 %
midterm exam, 25 % problem sets.
First problem set: due Tuesday, Sept. 22.
Problems 1-4 in appendix B.1 and Problems 1-4 in appendix B.2 of the
Second problem set: due Tuesday, Sept. 29.
Problems 1-3 in appendix B.3 of the book
Third problem set: due Thursday, Oct. 8.
Problem 4 in appendix B.3 and problems 1-4 in appendix B.4 of
Fourth problem set: due Tuesday, Oct. 20.
Problems 1 and 2 in appendix B.5 and problem 5 in appendix B.6 of
Fifth problem set: due Tuesday, Oct. 27.
Problems 1-3 in appendix B.6 of the book. Problem 4 in
appendix B.6 optional.
Sixth problem set: due Tuesday, Nov. 10.
Problems 1-4 in appendix B.7 of the book.
Seventh problem set: due Tuesday, Nov. 17.
Problem 1 in appendix B.8 of the book.
Eighth problem set: due Tuesday, Nov. 24.
Problems 1-3 in appendix B.9 of the book.
Ninth problem set: due Tuesday, Dec. 8.
Problems 1-4 in appendix B.10 of the book.
Tentative Schedule of Lectures
Chapter numbers correspond to the course textbook, Quantum
Theory, Groups and Representations.
September 8: Introduction and overview (Chapter 1)
September 10: The group U(1) and charge (Chapter 2)
September 15: Two-state systems and spin 1/2 (Chapter 3)
September 17: Linear algebra review, orthogonal and unitary groups
September 22: Lie algebras and Lie algebra representations
September 24: Rotations and spin in 3 and 4 dimensions (Chapter 6)
September 29: The spin 1/2 particle in a magnetic field (Chapter
October 1: Representations of SU(2) and SO(3) (Chapter 8)
October 6 : Tensor products (Chapter 9)
October 8: Review
October 13: Midterm exam (material through Chapter 9 of the notes)
October 15: Momentum and the free particle (Chapter 10)
October 20: Fourier analysis and the free particle (Chapter 11)
October 22: Position and the free particle (Chapter 12)
October 27: The Heisenberg group and the Schrödinger
representation (Chapter 13)
October 29: The Poisson bracket and symplectic geometry (Chapter
November 5: Quadratic polynomials and the symplectic group
November 10: Hamiltonian vector fields and the moment map (Chapter
November 12: Quantization (Chapter 17)
November 17: Semi-direct products (Chapter 18)
November 19: Quantum free particles and representations of the
Euclidean group (Chapter 19)
November 24: Representations of semi-direct products (Chapter 20)
December 1: Central potentials and the hydrogen atom (Chapter 21)
December 3: The harmonic oscillator (Chapter 22)
December 8: Coherent states (Chapter 23)
December 10: Review
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.
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
Groups and Symmetries: From Finite
Groups to Lie Groups, Yvette Kossmann-Schwarzbach
Introduction to Groups and Representations, Brian C. Hall
Lie groups, Lie algebras and
representations, Brian C. Hall
For more about Fourier analysis, see notes
from my Spring 2020 Fourier analysis class.
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