Applied Mathematics: Hilbert Spaces and Quantum Mechanics
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| David Hilbert |
Erwin Schroedinger |
Albert Einstein |
Images from: http://th.physik.uni-frankfurt.de/~jr/physlist.html.
The East Tennessee State University
Department of Mathematics will offer
a graduate level class in Applied Mathematics 1 (MATH 5610) which will
concentrate on Hilbert spaces and their application to
quantum mechanics.
The class will be
during the Fall 1998 term (August 31 - December 4), will meet
in Sam Wilson Hall, room 209 at 11:30-12:25 a.m.
Monday, Wednesday, and Friday, and will be taught by
Dr. Robert
Gardner.
The prerequisite for the class is a working knowledge of vector spaces and
elementary differential equations (as covered in Linear Algebra [MATH 2250]
and Differential Equations [MATH 3200]). A background in physics will be
useful, but there is no formal physics prerequisite.
The two texts to be used are
"Introduction to Hilbert Spaces with Applications"
by Lokenath Debnath and Piotr Mikusinski, copyright 1990 by
Academic Press
(ISBN 0-12-208435-7) and
"Quantum Mechanics and Experience"
by David Albert, available from
Harvard University Press (ISBN 0-674-74113-7).
Topics to be covered include:
- Normed Vector Spaces
Vector spaces, dimension, norms, completeness, Banach spaces, Contraction
Mapping Theorem.
- Hilbert Spaces - Properties and Geometry
Inner product spaces, Hilbert spaces, orthonormal systems, Fourier
series, linear functionals, bases, seperable Hilbert spaces.
- Hilbert Spaces - Linear Operators
Self-adjoint operators, unitary operators, projection operators, compact
operators, eigenvalues, Spectral Theorem.
- Quantum Mechanics
Postulates of quantum mechanics, Heisenberg
Uncertainty Principle, Schrodinger equation, linear harmonic
oscillators, angular momentum operators.
(Time permitting, existence and uniqueness theorems for ordinary
differential equations will be explored.) Each of these topics are
covered in the Debnath and Mikusinksi text. Additional topics to be
discussed are : superposition, von Neumann's formulation, Bohm's Theory,
Bell's Theorem, the Einstein-Podolsky-Rosen argument, and the
Kochen-Healy-Dieks interpretation (each covered in the Albert book).
Grades will be based on performance on homework, three in-class tests and
(depending on class size) in-class presentations. As a graduate level
class, there will be an emphasis on theorem and proof. Undergraduate math
and physics majors interested in a class covering this material, but at a
lower level, should contact
Dr. Robert
Gardner.
This course is the second of two planned courses offered by the ETSU Math
Department which deal in a mathematically rigorous way with topics of
modern physics (the first is
Differential Geometry,
which concentrated on special and general relativity, was offered during
Summer 1998, Term II).
Werner Heisenberg
(From http://th.physik.uni-frankfurt.de/~jr/gif/phys/heisenb2.jpg.)
Return to
Bob Gardner's home page