The spectrum of a chaotic quantum system was first suggested by Eugene P. Wigner, another early master of quantum mechanics. Wigner observed, as had many others, that nuclear physics does not possess the safe underpinnings of atomic and molecular physics: the origin of the nuclear force is still not clearly understood. He therefore asked whether the statistical properties of nuclear spectra could be derived from the assumption that many parameters in the problem have definite, but unknown values. This rather vague starting point allowed him to find the most probable formula for the distribution. Oriol Bohigas and Marie-Joya Giannoni of the Institute of Nuclear Physics in Orsay, France, first pointed out that Wigner's distribution happens is be exactly what is found for the spectrum of a chaotic dynamic system.
Chaos does not seem to limit itself to the distribution of quantum energy levels, however, it even appears to work its way into the wavelike nature of the quantum world. The position of the electron in the hydrogen atom is described by a wave pattern. The electron cannot be pinpointed in space; it is a cloudlike smear hovering near the proton. Associated with each allowed energy level is a stationary state, which is a wave pattern that does not change with time. A stationary state corresponds quite closely to the vibrational pattern of a membrane that is stretched over a rigid frame, such as a drum.
The stationary states of a chaotic system have surprisingly interesting structure, as demonstrated in the early 1980s by Eric Heller of the University of Washington. He and his students calculated a series of stationary states for a two-dimensional cavity in the shape of a stadium. The corresponding problem in classical mechanics was known to be chaotic, for a typical trajectory quickly covers most of the available ground quite evenly. Such behavior suggests that the stationary states might also look random, as if they had been designed without rhyme or reason. In contrast. Heller discovered that most stationary states are concentrated around narrow channels that form simple shapes inside the stadium, and he called these channels "scars" [see illustration]. Similar structure can also be found in the stationary states of a hydrogen atom in a strong magnetic field [see illustration]. The smoothness of the quantum wave forms is preserved from point to point, but when one steps back to view the whole picture, the fingerprint of chaos emerges.
It is possible to connect the chaotic signature of the energy spectrum to ordinary classical mechanics. A clue to the prescription is provided in Einstein's 1917 paper, He examined the phase space of a regular system from box R and described it geometrically as filled with surfaces in the shape of a donut; the motion of the system corresponds to the trajectory of a point over the surface of a particular donut. The trajectory winds its way around the surface of the donut in a regular manner, but it does not necessarily close on itself.
In Einstein's picture, the application of Bohr's correspondence principle to find the energy levels of the analogous quantum mechanical system is simple. The only trajectories that can occur in nature are those in which the cross section of the donut encloses an area equal to an integral multiple of Planck's constant, h (2π times the fundamental quantum of angular momentum having the units of momentum multiplied by length). It turns out that the integral multiple is precisely the number that specifies the corresponding energy level in the quantum system.
Unfortunately as Einstein clearly saw, his method cannot be applied if the system is chaotic, for the trajectory does not lie on a donut and there is no natural area to enclose an integral multiple of Planck's constant. A new approach must be sought to explain the distribution of quantum mechanical energy levels in terms of the chaotic orbits of classical mechanics.
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11 Comments
Add CommentI long for the good old days when intelligent people dominated the news. The only way to get attention these days is to forecast doom or blow yourself up!
Reply | Report Abuse | Link to thisAm I being thick? The text refers to a number of crucial figures, where are they?
Reply | Report Abuse | Link to thisTying this all to the zeta function seems to always be where such discussions of quantum (and other) chaos end. So many things seem to be built of characters which resemble the prime numbers - doesn't anyone notice the semantics of it all? When one asks the question " What exactly do I mean by prime and composite?" - one begins to realize that mathematics is "just another language".
Reply | Report Abuse | Link to thisWHERE ARE THE FIGURES???
Reply | Report Abuse | Link to thisSorry about the lack of figures, folks. We didn't have them handy and thought it more important to get the article out than to wait for the images.
Reply | Report Abuse | Link to thisThe article is excellent. However, the lack of figures illustrating/discussing the text makes it much less valuable than it should be. Surely SciAm should have been able to get hold of those figures by now? Please do publish them and let us know that you have done so. Thanks, -- GSC
Reply | Report Abuse | Link to thisIn arithmetic, operations are interchangeable if they are commutative. Nest non-commutative operations in program loops and results will be truncated with rounding errors, because decimal numbers are not reals. So such simple iterative loops cause results to go haywire within a few dozen repetitions. You can write computer programs to calculate as we do long-hand to great precision, but this increases execution times exponentially. No doubt to calculate quantum phases, computers are used, but how can we write programs which are not inherently chaotic, in view of the fact that we cannot know the position and velocity of a sub-atomic particle. Chaos theory applies to calculations as much as to physics. This article begs the question?
Reply | Report Abuse | Link to thisIt's excellent
Reply | Report Abuse | Link to thisThere are so many things to be understood from that text that I'm sure I don't understand the tenth of it.
Reply | Report Abuse | Link to thisBut quantum chaos seems to be a link between quamtum and classical mechanics. I am a determinist, I think that the probabilistic description of current quantum mechanics is just a statistical view of something that is in fact much more complex - a new mechanics to discover ? Anyway, I'm sure that quantum chaos has a role to play in all of this, I guess the future greatest progress in fundamental physics will have to do something with quantum chaos.
Consider the answer lays in the faqct that we need to look at the Universe at a much smaller scale.
Reply | Report Abuse | Link to thisWhat if the electron itself was not a fundamental particle but made up from 1.23 x 10 to the power of 20, such particles. Now everything makes a lot more sense it is not a probability density distribution -it is a cloud. Moreover, a cloud is almost exactly what you see when you look at the electron experimentally and mathematically.
You get quiter a lot of determinim back into your equations, but you haqve to remeber at that scale space-time is made nof the same harmonic quintessence. bottom mline is nyou will almost always get some chaos coming in - hence Heisenberg's uncertainty.
The bonus is that you can now deirive things like E=mc2 from first principles. See: The formulation od harmonic quintessence and a fundamental enenrgy equivalence equation. See Physics Essays 23: 311-319.
You can find the figures at http://www.dhushara.com/book/quantcos/qchao/quantc.htm I'm not sure why they haven't been included...?
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