MAT 649, Spring 2001, Topics in Mathematical Physics
RIGOROUS QUANTUM MECHANICS
Lecturer: Marco Lenci. Office:
Math, 5D-148. Tel.: xxx-xxxx. E-mail: xxxx@math.sunysb.edu.
Office hours: by appointment.
Time and place: MW 7:00--8:20, Physics P-124.
Course description:
This is intended to be a fairly organic course in basic (i.e.,
non-relativistic) Quantum Mechanics (which the typical mathematician
is not usually exposed to), but with the methods of rigorous
mathematics (which the typical physicist is not usually exposed to).
The differences between this and a standard first course in
Quantum Mechanics are mainly two: a different spin in the choice and
juxtaposition of the topics, and the presentation of the subject
within the scope of a mathematically self-contained theory. (To what
extent this theory has to do with reality is an extremely interesting
question which would be nice to discuss briefly, if time permits.)
A course like this might interest the student with an
inclination for Mathematical Physics (whatever that is). In
particular, it could be the starting point for further studies in
Semiclassical Analysis, Quantization, Quantum Chaos, Foundations of
Quantum Mechanics, Quantum Field Theory, etc.
(Tentative) Syllabus:
-
Basic concepts of Quantum Mechanics
- (Short) hystorical background
- Spectral theory of self-adjoint operators
- Axioms of Quantum Mechanics and their consequences (Uncertainty
Principle, etc.)
- Schroedinger representation and Schroedinger equation
- Examples (harmonic oscillator, potential well, free particle, etc.)
- Generalized eigenfunctions
-
Properties of the Schroedinger equation
- Self-adjointness (Kato's theorem, ...)
- 1D Schroedinger operators with increasing potential
- Discrete spectrum and localization
- Periodic potentials?
- Examples of multidimensional Schroedinger equation (hydrogen atom?)
- Weyl asymptotics?
-
Quantization
- Weyl correspondence
- Heisenberg group and Stone-Von Neumann theorem
- Pseudodifferential operators and symbols?
-
If we have time (and endurance), one topic among
- Second quantization
- Quantum Chaos
- Foundational questions
Texts:
Understandably, there is a huge literature in this subject. However,
there aren't many books that adopt the cut of this course. I found a
suitable one just recently (which means I'm not familiar with it yet):
- F. A. Berezin and M. A. Shubin, The Schroedinger equation,
Kluwer
so this is the officially recommended text. On the other hand, it is
unavoidable (and beneficial!) that we pick more or less heavily from
other books as well. I list some in order of usage:
- J. Weidmann, Linear operators in Hilbert spaces,
Springer
- L. D. Landau and E. M. Lifshitz, Quantum mechanics,
non-relativistic theory, Addison-Wesley
- G. .B. Folland, Harmonic analysis in phase space,
Princeton
- E. Merzbacher, Quantum mechanics, Wiley
- M. Reed and B. Simon, Methods of modern mathematical physics,
Academic
- M. A. Shubin, Pseudodifferential operators and spectral
theory, Springer
For the more advanced topics
- F. A. Berezin, The method of second quantization,
Academic
- M. C. Gutzwiller, Choas in classical and quantum mechanics,
Springer
- N. E. Hurt, Quantum chaos and mesoscopic systems, Kluwer
- J. S. Bell, Speakable and unspeakable in quantum mechanics,
Cambridge
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Last updated: Apr 2, 2001 (adapted to ML's Stevens
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