Ma 148 c: Quantum Physics for Mathematicians
Spring 2011, Caltech Math Department, Tuesday-Thursday 1:00-2:30 pm.
Instructor:
Matilde Marcolli
Brief Course Description
We will be covering topics from Quantum Mechanics (and, time
permitting, some Quantum Field Theory) from a mathematical
perspective.
Prerequisites:
Previous exposure to some quantum physics and mathematical
analysis can be useful, but are not strictly required, as
the class with try as much as possible to be self-contained.
Textbook
Leon A. Takhtajan, "Quantum mechanics for mathematicians", American
Mathematical Society, 2008.
Additional suggested readings
- L.D.Faddeev, O.A.Yakubovskii, "Lectures on quantum mechanics
for mathematics students", American Mathematical Society, 2009.
- Floyd Williams, "Topics in Quantum Mechanics", Birkhauser, 2003.
Further reading material will be added to the list as the class
progresses.
Grading policy
Given the large number of registered students, the class is
offered P/F only.
There is a TA, Branimic Cacic, who will set up
recitation and discussion sessions for the class.
Grading will be based on participation in class.
Outline of classes and notes
- Tuesday March 29: smooth manifolds and vector bundles, tensors;
Lagrangian mechanics, configuration space, least action principle,
Euler-Lagrange equations Notes
- Thursday March 31: Noether's theorem; differential forms,
symplectic manifolds; Legendre transform, phase space,
Hamiltonian mechanics, Hamilton equations Notes 1, Notes 2
- Tuesday April 5: Poisson structures; observables, states and
measurement in classical mechanics; Hamilton and Liouville's points
of view on mechanics
- Thursday April 7: Quantum Mechanics: Hilbert spaces and
operators (bounded, self-adjoint, compact, positive...)
- Tuesday April 12: Observables and states: Heisenberg and Schroedinger's
points of view on quantum mechanics; pure states and Schroedinger equation
- Thursday April 14: Quantization of classical system: position and momentum
representations
- Tuesday April 19: Weyl quantization and the Weyl transform
- Thursday April 21: Deformation quantization and the Weyl quantization
- Tuesday April 26: Deformation quantization: Hochschild cohomology
and deformations of algebras
- Thursday April 28: Gelfand triples (rigged Hilbert spaces): generalized
eigenfunctions and spectral theorem in rigged spaces
- Tuesday May 3: Schroedinger equation and hypergeometric equations:
canonical form and families of polynomial solutions
- Thursday May 5: Schroedinger equation and hypergeometric
equations, the harmonic oscillator
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