Description : our goal is to explore some beautiful and fundamental mathematics in the physical context of quantum mechanics. The focus will be on analytic techniques (examples: operators on Hilbert space, spectral theory, the Fourier transform, the stationary phase method, calculus of variations) applied to quantum mechanical problems. Of course, these methods are important in a wide variety of fields beyond quantum theory (mathematical physics more generally, differential equations, probability, applied mathematics, geometry, etc.)
Rough Outline :
• some physical background: wave functions and the Schr¨
• Hilbert spaces, (unbounded) self-adjoint operators, and solving Schr¨ equation
• the Fourier transform, observables, the uncertainty principle, and quantization
• some spectral theory, and applications to quantum dynamics
• further topic(s) chosen from among
– many-body theory, and Bose-Einstein condensation
– scattering theory
– quantum information theory
Useful Background : some experience with analysis (say at the level of Math
420/421 or equivalent) would be useful, though basic definitions and properties will be given. Some physics would be an asset, but is not required. Math 511 is particularly good preparation. These are just guidelines (please talk to the instructor if unsure).
Instructor: Stephen Gustafson, Math 115, phone 604-822-3138, gustaf@math.ubc.ca.
References : we will follow closely parts of
· S. Gustafson, I.M. Sigal Mathematical Concepts of Quantum Mechanics (Springer)
An additional list of references for both mathematical and physical background will be maintained on the course website.
Grading: is based on regular homework assignments, plus (possibly) an oral presentation
Aug. 28, 2011