I wrote this article in the summer of 1998 (the third year of my PhD),
it was published in the
School of Physics and Astronomy 1999 newsletter.
PhD Experiences of Chaos and Uncertainty Principles
"A PhD in theoretical physics - nice idea."
"Quantum chaos, two of the biggest buzz-words in physics - sounds cool."
In a sound-bite world, another career decision is made. I had joined that
select group of physicists who never get to clamber over apparatus, tweaking
knobs and cursing vacuum pumps. There is no escape from my desk, with its
scree of paper - each sheet covered with spidery writing, mathematical symbols
with more subscripts than nature ever intended. I have no lab to sneak off to.
The nearest thing I have to a high voltage power supply is a kettle.
I once heard a rumour that more theoretical physicist are working on quantum
chaos than any other field of research. I've mentioned it twice, so I hope
by now you are at least a little curious about quantum chaos.
Many things are chaotic. The weather is the most famous example,however there
are many much simpler systems which are also chaotic. Take a system obeying
good old-fashioned Newtonian mechanics, a ball on a billiard table. A ball on
an idealised rectangular billiard table is not chaotic. (Idealised billiard
tables have no pockets!) However if you make the billiard table almost any
shape other then rectangular or circular, then it will be chaotic. A nice
example is the stadium billiard table which looks like a stadium or running
track viewed from above.
The initial position and momentum of the ball determine the path that it
follows as it bounces round the table. Give the ball ever so slightly
different initial conditions, it will follow a different path. If the system
is chaotic, the two paths will diverge from each other exponentially quickly.
In non-chaotic systems, this does not happen. So, in a chaotic system, even
if the differences between the initial conditions of the two paths are too
small to be measured, these small differences are amplified until the two
paths are utterly different.
They don't give degrees away to just anyone, (these days you need to have
a thousand quid burning a hole in your pocket), so you will have guessed that
I will now introduce quantum mechanics into the problem.
If the object bouncing about on the billiard table is a quantum mechanical
particle, for example an electron, it will still behave like the classical
ball so long as the wavelength of the electron is negligible compared with the
size of the billiard table. However as we reduce the size of the billiard
table, classical mechanics will no longer be a good approximation of the
quantum mechanical behaviour of the particle.
So what does the quantum mechanics of one of these chaotic billiards look like?
Well, pretty much like any other quantum system. Pick your favourite; waves
in a box, the hydrogen atom. They all have eigenstates which do not vary with
time and have a well defined energy. The quantum mechanics of these chaotic
billiards is no different.
This raises a problem. Why is the classical behaviour of chaotic systems so
different from non-chaotic systems, when the underlying quantum mechanics is
so similar? A lot of work has been done to try and answer this question.
However even the most successful theories have not yet succeeded.
There is a lot of debate about how to improve these theories, unfortunately it
is not obvious how to estimate, let alone improve, their accuracy. My research
involves applying these techniques to a system of randomly placed scatterers.
This system has been very well studied, because it is a good model of a metal
or superconductor with impurities in it. There are a number of powerful
theories which enable us to know almost everything about these systems.
Therefore it is an ideal place to test out the quantum chaos theories, find out
their accuracy and hopefully improve them.
So there is the Big Picture, and I am a big picture person, the Jungian
personality test I did the other day said so. On a day-to-day basis, life in
our office is sarcasm, stolen pens, and herbal tea. Like all good Marxists,
we only drink herbal tea, because proper tea is theft. I have been lost in an
algebra jungle for the last month or so, however I am slowly hacking my way
towards civilisation. Outside the sun is shining, undergrads have finished
their exams and the World Cup is on the TV. Motivation is something that only
happens to other people. Still the sooner I get through the algebra, the
sooner I can get back to the big picture. Now where is my pen?