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

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.

Problem Set 1 (Due Tuesday, September 16)
Problem Set 2 (Due Tuesday, September 30)
Problem Set 3 (Due Tuesday, October 7)
Problem Set 4 (Due Tuesday, October 21)
Problem Set 5 (Due Thursday, November 6)
Problem Set 6 (Due Tuesday, November 18)
Problem Set 7 (Due Tuesday, December 2)
Final problem set (Due at final exam)

Problem sets from the last version of this course are available here.

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

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: Representations of semi-direct products (Chapter 17)
Tuesday, November 25:  Quantum free particles and representations of the Euclidean group (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