Introduction to Quantum Mechanics:
Mathematics GU4391 (spring 2022)
Peter Woit (woit@math.columbia.edu)
Mathematics 421
Tuesday and Thursday 2:40-3:55pm
407 Mathematics
First four classes currently planned to be on Zoom, links
available at the Courseworks site for this class.
For the first two weeks I won't schedule office hours, but will be
available in Mathematics 421 most times, feel free to stop by, or
email and set up a meeting time. Also email if you would like
to schedule a meeting on Zoom.
Teaching Assistant: Zoe Himwich (zmh2110@columbia.edu)
Course Summary and Prerequisites
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 a conventional first physics course on the subject
will be covered, but with much greater attention to the mathematical
ideas behind the standard formalism and usual calculational
techniques.
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.
Lecture Notes/Book/Videos
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 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.
Last time I taught the class, it was online-only because of COVID
and videos are available
on Youtube.
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 Thursday, May 12,
1:10-4pm. 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, Feb. 1 .
Problems 1-4 in appendix B.1 and Problems 1-4 in appendix B.2 of the
book
Second problem set: due Thursday, Feb. 10.
Problems 1-3 in appendix B.3 of the book
Third problem set: due Tuesday, Feb. 15.
Problem 4 in appendix B.3
Fourth problem set: due Tuesday, Feb. 22.
Problems 1-4 in appendix B.4 and problem 1 in appendix B.5 of the
book
Fifth problem set: due Tuesday, March 8.
Problems 1-2 and 5 in appendix B.6 of the book.
Sixth problem set: due Tuesday, March 22.
Problem 3 in appendix B.6 of the book. Problem 4 in appendix
B.6 optional.
Seventh problem set: due Tuesday, April 5.
Problems 1-4 in appendix B.7 of the book.
Eighth problem set: due Tuesday, April 12.
Problem 1 in appendix B.8 of the book.
Ninth problem set: due Tuesday, April 19.
Problems 1-3 in appendix B.9 of the book.
Tenth problem set: due Thursday, April 28.
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.
Tuesday,
January 18: Introduction and overview (Chapter 1)
Thursday,
January 20: The group U(1) and charge (Chapter 2)
Tuesday,
January 25: Two-state systems and spin 1/2 (Chapter 3)
Thursday,
January 27: Linear algebra review, orthogonal and unitary groups
(Chapter 4)
Tuesday,
February 1: Lie algebras and Lie algebra representations (Chapter
5)
Thursday,
February 3: Rotations and spin in 3 and 4 dimensions (Chapter 6)
Tuesday,
February 8: The spin 1/2 particle in a magnetic field (Chapter 7)
Thursday,
February 10: Representations of SU(2) and SO(3) (Chapter 8)
Tuesday,
February 15 : Tensor products (Chapter 9)
Thursday,
February 17: Review
Tuesday,
February 22: Midterm exam (material through Chapter 9 of the
notes)
Thursday,
February 24: Momentum and the free particle (Chapter 10)
Tuesday,
March 1: Fourier analysis and the free particle (Chapter 11)
Thursday,
March 3: Position and the free particle (Chapter 12)
Tuesday,
March 8: The propagator for a free particle (Chapter 12)
Thursday,
March 10: The Heisenberg group and the Schrödinger representation
(Chapter 13)
Tuesday, March 22: The Poisson bracket and symplectic geometry
(Chapter 14)
Thursday, March 24: Quadratic polynomials and the symplectic group
(Chapter 16)
Tuesday, March 29: More on quadratic polynomials and the
symplectic group (Chapter 16)
Thursday,
March 31: Hamiltonian vector fields and the moment map (Chapter
15)
Tuesday, April 5: Quantization (Chapter 17)
Thursday, April 7: The Euclidean group (Chapter 18)
Tuesday, April 12: Quantum free particles and representations of
the Euclidean group (Chapter 19)
Thursday, April 14: Central potentials and the hydrogen atom
(Chapter 21)
Tuesday, April 19: More on the hydrogen atom
Thursday, April 21: The harmonic oscillator (Chapter 22)
Tuesday, April 26: More on the harmonic oscillator
Thursday, April 28: Review
Final exam is scheduled for Thursday, May 12, 1:10-4pm
Other Textbooks
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(The following is from when I
last taught the course, will be updated soon for the spring 2022
version). 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
For more about Fourier analysis, see notes
from my Spring 2020 Fourier analysis class.
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
Introduction to Quantum Mechanics, Spring
2021: Math GU4392