Theory of quantum computation.
Major efforts are currently underway to build a new kind of computer, which
carries out calculations using quantum rather than classical operations.
If such a computer could be built, its ability to exploit the inherent
parallelism in quantum mechanical interference would allow it to do certain
calculations much more efficiently than a classical computer. The
best-known example is the quantum algorithm for factoring large numbers,
which is exponentially faster than all known classical algorithms. Since
modern encryption techniques rely on the inefficiency of the classical
factoring algorithm for their security, such a computer could potentially
break the most secure codes in use today. Using computer simulations and
analytic calculations, the student will investigate the theoretical
possibilities of quantum computing, focusing on what speedups are available
(and which are not) for fundamental problems in computer science.
Solitons and oscillons.
With the exception of electromagnetism, the fundamental forces of nature
are all nonlinear -- they do not obey the superposition principle. (And
even electromagnetism is nonlinear when quantum mechanics is included.)
One striking feature of many nonlinear systems is the appearance of
localized lumps of fields, held together by their own self-interactions.
These lumps can be static (solitons) or oscillatory (oscillons). Such
objects cannot occur in a linear theory, where waves simply disperse (like
ripples in a pond or the beam of a flashlight). Solitons and oscillons can
play a crucial role in physical systems ranging from superconductors to the
early universe, but relatively little is known about them except in a few
special cases. The power of modern computation, however, offers
significant opportunities to change this situation. The student will
investigate the properties of solitons or oscillons in a particular
nonlinear problem chosen commensurate with background and interest.
Topics in elementary particle physics.
The student will investigate aspects of the modern theory of elementary
particles. Possible topics could include the quark model, symmetries and
symmetry breaking, unification of forces, gauge theories, or other topics
depending on background and interest.
General relativity and the cosmological constant.
Einstein's theory of general relativity is more than 80 years old, but it
remains the best description of gravity we have. Starting with its
mathematical foundations in Riemannian geometry, the student will learn the
theory of general relativity. The goal will be to understand the
cosmological constant, which is an extra term originally introduced into
the equations of general relativity by Einstein. He was trying to explain
how the universe could be static, as it was believed to be at the time,
instead of expanding, as predicted by his theory without the cosmological
constant. Once the expansion of the universe was observed, Einstein called
his introduction of the cosmological constant his greatest blunder. But
recent observations of distant supernovae have demonstrated that the
cosmological constant term does indeed exist after all (although it is very
small). Explaining its origin is probably the biggest unsolved mystery
in particle physics today.
The Casimir effect in quantum field theory.
When quantum mechanics is combined with special relativity, it is no longer
possible to consider only one particle at a time, as we do in ordinary
quantum mechanics, or to consider continuous fields, as we do in
electromagnetism. Instead, we are forced to combine these two pictures
into a quantum field theory by building the continuous field out of
discrete units of each possible mode of oscillation of the field. For
example, the electromagnetic field is constructed out of a given number of
photons for each possible wave vector. This project will investigate one
surprising result from quantum field theory, the Casimir effect. It turns
out that there is a small (but measurable) energy associated with empty
space between uncharged conducting plates, which is obtained by summing
contributions that arise from every mode of excitation of the system --
which exist even when no modes are excited. Since each mode of oscillation
is described by the equations of quantum mechanics, computing this sum
requires the use of numerical and analytic techniques extending those
discussed in PH202 and PH401. Recent experimental advances have made it
possible to measure this effect experimentally; it is now of significant
interest in nanotechnology applications. The student will learn concepts
from advanced quantum mechanics, such as scattering phase shifts, and apply
them to Casimir effect research.
Visual processing in the eye.
In quantum mechanics, the uncertainly principle limits the precision with
which one can simultaneously measure a particle's position and momentum.
Photoreceptors in your eye confront the same tradeoff; for example, when
looking at a surface, you want to be able to detect both edges (the analog
of position) and texture (the analog of momentum). In quantum mechanics,
physicists discovered that coherent states provide the optimal tradeoff
between these two competing goals. Evolution discovered these states as
well, and uses them to optimize the performance of photoreceptors in the
retina. A student with some familiarity with computer programming could
carry out some simple experiments to demonstrate how to use the
mathematical tools from quantum mechanics to understand visual recognition
systems.
Physics and automatic speech recognition.
Modern automatic speech recognition systems demonstrate the power of using
statistical techniques to model complex phenomena. Pattern-matching
systems, such as hidden Markov models and neural networks, allow one to
incorporate statistical models for a wide variety of acoustic and syntactic
data. Physics concepts ranging from wave mechanics to entropy are useful
in this process. A student comfortable with computer programming can carry
out experiments using publicly available speech recognition systems and
look for improvements in accuracy and performance.