There is a fascinating article about quantum mechanics at Scientific

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