Introduction to Quantum Mechanics:
Mathematics W4391 (fall 2014)
Tuesday and Thursday 4:10-5:25pm
Mathematics 307
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 W4392) 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.
Lecture Notes
Note: The lecture notes for the course have been turned
into a book, available here.
I've removed the old lecture notes and problem sets, since
better versions of this material are incorporated in the book.
A very detailed set of notes for this course is under
development, with the latest version always available here.
During the fall semester I expect to cover roughly the material in
the first 22 chapters of the notes. 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
There will be problem sets due roughly every other week, a midterm
and a final exam.
Exams
There will be a midterm exam Tuesday, October 7. Use of notes is
allowed.
The final exam is scheduled for Tuesday, December 16,
4:10-7pm. Use of notes is allowed.
Syllabus
Tuesday, September 2: Introduction and overview (Chapter 1)
Thursday, September 4: More overview
Tuesday, September 9: The group U(1) and Charge (Chapter 2)
Thursday, September 11: More on U(1), two-state systems and
spin 1/2 (Chapter 3)
Tuesday, September 16: Two-state systems and spin 1/2 (Chapter 3)
Thursday, September 18: Linear algebra review, orthogonal and
unitary groups (Chapter 4)
Tuesday, September 23: Lie algebras and Lie algebra representations
(Chapter 5)
Thursday, September 25: More on Lie algebra representations,
rotations in 3 and 4 dimensions (Chapter 6)
Tuesday, September 30: More on rotations in 3 and 4 dimensions
Thursday, October 2: The spin 1/2 particle in a magnetic field
(Chapter 7)
Tuesday, October 7: Midterm exam (material through Chapter 7 of the
notes)
Thursday, October 9: Representations of SU(2) and SO(3) (Chapter 8)
Tuesday, October 14: More on representations of SU(2) and SO(3),
Tensor products (Chapter 9)
Thursday, October 16: Tensor products (Chapter 9)
Tuesday, October 21: More on tensor products. Energy and momentum:
the free particle (Chapter 10)
Thursday, October 23: More on the free particle. Basics of
Fourier analysis (Chapter 10)
Tuesday, October 28: The Heisenberg group and the Schrodinger
representation (Chapter 11)
Thursday, October 30: The Poisson bracket and symplectic geometry
(Chapter 12)
Thursday, November 6: The moment map (Chapter 13)
Tuesday, November 11: Quadratic polynomials and the symplectic group
(Chapter 14)
Thursday, November 13: Quantization (Chapter 15)
Tuesday, November 18: Semi-direct products (Chapter 16)
Thursday, November 20: Quantum free particles and representations of
the Euclidean group (Chapter 17)
Tuesday, November 25: Representations of semi-direct products
(Chapter 18)
Tuesday, December 2: Central potentials and the hydrogen atom
(Chapter 19)
Thursday, December 4: The harmonic oscillator (Chapter 20)
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.
The book
Linearity, Symmetry and Prediction
in the Hydrogen Atom, Stephanie Singer, Springer, 2005.
covers some of the material we will cover, especially the hydrogen
atom spectrum calculation, from a point of view similar to the one
of this course. It is available free on SpringerLink from
Columbia addresses at this
URL. At this site one can also purchase a printed copy
of the book for $24.95
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