American.
Here is a short abstract from that article, the original can be found
at http://www.sciam.com/article.cfm?id=quantum-chaos-subatomic-worlds
'In 1917 Albert Einstein wrote a paper that was completely ignored for
40 years. In it he raised a question that physicists have only,
recently begun asking themselves: What would classical chaos, which
lurks everywhere in our world, do to quantum mechanics, the theory
describing the atomic and subatomic worlds? The effects of classical
chaos, of course, have long been observed-Kepler knew about the motion
of the moon around the earth and Newton complained bitterly about the
phenomenon. At the end of the 19th century the American astronomer
William Hill demonstrated that the irregularity is the result entirely
of the gravitational pull of the sun. So thereafter, the great French
mathematician-astronomer-physicist Henri Poincaré surmised that the
moon's motion is only mild case of a congenital disease affecting
nearly everything. In the long run Poincaré realized, most dynamic
systems show no discernible regularity or repetitive pattern. The
behavior of even a simple system can depend so sensitively on its
initial conditions that the final outcome is uncertain.
At about the time of Poincaré's seminal work on classical chaos, Max
Planck started another revolution, which would lead to the modern
theory of quantum mechanics. The simple systems that Newton had
studied were investigated again, but this time on the atomic scale.
The quantum analogue of the humble pendulum is the laser; the flying
cannonballs of the atomic world consist of beams of protons or
electrons, and the rotating wheel is the spinning electron (the basis
of magnetic tapes). Even the solar system itself is mirrored in each
of the atoms found in the periodic table of the elements.
Perhaps the single most outstanding feature of the quantum world is
its smooth and wavelike nature. This feature leads to the question of
how chaos makes itself felt when moving from the classical world to
the quantum world. How can the extremely irregular character of
classical chaos be reconciled with the smooth and wavelike nature of
phenomena on the atomic scale? Does chaos exist in the quantum world'?
Preliminary work seems to show that it does. Chaos is found in the
distribution of energy levels of certain atomic systems; it even
appears to sneak into the wave patterns associated with those levels.
Chaos is also found when electrons scatter from small molecules. I
must emphasize, however, that the term "quantum chaos" serves more to
describe a conundrum than to define a well-posed problem.
Considering the following interpretation of the bigger picture may be
helpful in coming to grips with quantum chaos. All our theoretical
discussions of mechanics can be somewhat artificially divided into
three compartments [see illustration] although nature recognizes none
of these divisions.
Elementary classical mechanics falls in the first compartment. This
box contains all the nice, clean systems exhibiting simple and regular
behavior, and so I shall call it R, for regular. .Also contained in R
is an elaborate mathematical tool called perturbation theory which is
used to calculate the effects of small interactions and extraneous
disturbances, such as the influence of the sun on the moon's motion
around the earth. With the help of perturbation theory, a large part
of physics is understood nowadays as making relatively mild
modifications of regular systems. Reality though, is much more
complicated; chaotic systems lie outside the range of perturbation
theory and they constitute the second compartment.
Since the first detailed analyses of the systems of the second
compartment were done by Poincaré, I shall name this box P in his
honor. It is stuffed with the chaotic dynamic systems that are the
bread and butter of science. Among these systems are all the
fundamental problems of mechanics, starting with three, rather than
only two bodies interacting with one another, such as the earth, moon
and sun, or the three atoms in the water molecule, or the three quarks
in the proton.
Quantum mechanics, as it has been practiced for about 90 years,
belongs in the third compartment, called Q. After the pioneering work
of Planck, Einstein and Niels Bohr, quantum mechanics was given its
definitive form in four short years, starting in 1924. The seminal
work of Louis de Broglie, Werner Heisenberg, Erwin Schrödinger, Max
Born, Wolfgang Pauli and Paul Dirac has stood the test of the
laboratory without the slightest lapse. Miraculously. it provides
physics with a mathematical framework that, according to Dirac, has
yielded a deep understanding of "most of physics and all of chemistry"
Nevertheless, even though most physicists and chemists have learned
how to solve special problems in quantum mechanics, they have yet to
come to terms with the incredible subtleties of the field. These
subtleties are quite separate from the difficult, conceptual issues
having to do with the interpretation of quantum mechanics.
The three boxes R (classic, simple systems), P (classic chaotic
systems) and Q (quantum systems) are linked by several connections.
The connection between R and Q is known as Bohr's correspondence
principle. The correspondence principle claims, quite reasonably, that
classical mechanics must be contained in quantum mechanics in the
limit where objects become much larger than the size of atoms. The
main connection between R and P is the Kolmogorov-Arnold-Moser (KAM)
theorem. The KAM theorem provides a powerful tool for calculating how
much of the structure of a regular system survives when a small
perturbation is introduced, and the theorem can thus identify
perturbations that cause a regular system to undergo chaotic behavior.
Quantum chaos is concerned with establishing the relation between
boxes P (chaotic systems) and Q (quantum systems). In establishing
this relation, it is useful to introduce a concept called phase space.
Quite amazingly this concept, which is now so widely exploited by
experts in the field of dynamic systems, dates back to Newton.
The notion of phase space can be found in Newton's mathematical
Principles of Natural Philosophy published in 1687. In the second
definition of the first chapter, entitled "Definitions", Newton states
(as translated from the original Latin in 1729): "The quantity of
motion is the measure of the same, arising from the velocity and
quantity of matter conjointly." In modern English this means that for
every object there is a quantity, called momentum, which is the
product of the mass and velocity of the object.
Newton gives his laws of motion in the second chapter, entitled
"Axioms, or Laws of motion." The second law says that the change of
motion is proportional to the motive force impressed. Newton relates
the force to the change of momentum (not to the acceleration as most
textbooks do).
Momentum is actually one of two quantities that, taken together, yield
the complete information about a dynamic system at any instant. The
other quantity is simply position, which determines the strength and
direction of the force. Newton's insight into the dual nature of
momentum and position was put on firmer ground some 130 years later by
two mathematicians, William Rowan Hamilton and Karl Gustav-Jacob
Jacobi. The pairing of momentum and position is no longer viewed in
the good old Euclidean space or three dimensions; instead it is viewed
in phase space, which has six dimensions, three dimensions for
position and three for momentum.
Continue reading the article at http://www.sciam.com/article.cfm?id=quantum-chaos-subatomic-worlds&page=3
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