#
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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