(For those who came last year and remember that the meeting was also on
December 23rd, please note this is purely coincidental, future meetings
may be held on different dates.)
The idea of the workshops is to create a forum for workers in applied
mathematics, especially younger faculty and students, to get to know other
members of the community, and promote discussion and collaboration.
This year's workshop will take place in the lecture hall on the first floor of
the Economics building (building 504) on Bar-Ilan's main campus. See
The Campus Map. For those arriving
by car, there are parking lots outside, but close to, the campus
(the big white things on the map next to building 1401). A large number of
bus routes serve the university.
Mathematical Challenges from Classical and Quantum Dynamics
David J. Tannor (Department of Chemical Physics,
Weizmann Institute of Science)
With the exception of a few instances where relativistic effects
are important, a single equation - the time-dependent Schrodinger
equation - describes all of atomic and molecular physics. The
problem is that formally, this wave equation has to be solved for
all the electrons and nuclei in the atom or molecule. For a small
molecule such as ozone (three oxygen atoms) there are already 24
electrons and three nuclei, for a total of 3 x 27 = 81 degrees of
freedom. For the smallest peptide molecule, glycine (a single
link in a DNA chain that typically extends for many thousands of
peptides) there are already 3 x 50 =150 degrees of freedom.
Solving a wave equation exactly in so many degrees of freedom
is out of the question. One of the main challenges for theoretical
chemistry is find accurate and efficient approximate solutions to
wave equations in so many degrees of freedom.
I belong to a group
of theoretical chemists who assume that the Schrodinger equation
for the electrons has already been solved. The remaining part of
the problem is the dynamics of the relative motion of the nuclei
in the presence of the electronic force field, so-called chemical
reaction dynamics. There are only 9 nuclear degrees of freedom
in ozone, and only 30 in glycine, but solving the wave equation
is still a formidable or impossible task.This talk will focus on
three aspects of the above problem. 1) Several classes of methods
used to solve the time dependent Schrodinger equation for the
nuclei exactly in up to 6 degrees of freedom will be described.
Recent work on pseudospectral methods that avoid an underlying
direct product basis will be presented (collaboration with Ilan
Degani and Jeremy Schiff). 2) A large literature devoted to
using classical mechanics reliably to approximate quantum
mechanics will be summarized. Recent ideas for using classical
mechanics to create pseudospectral representations that avoid
an underlying direct product basis will be described
(collaboration with Yair Goldfarb). I will also discuss
methods for using classical mechanics to propagate the time
dependent Schrodinger equation. 3) Finally, I will report
on ongoing efforts in my group to use optimal control theory
to design specially shaped laser pulses to control photochemical
reactions. Here, the wave character of the light and the
wave nature of the matter must be solved within a single coherent
framework. The mechanism for control is via constructive or
destructive interference, representing an interesting paradigm
for a control problem (collaboration with Shlomo Sklarz)
