Mathematicians settle 30-year-old resonance controversy

November 3, 2014 by Lisa Zyga feature
resonances
Mathematicians have used a method based on interval arithmetic to locate resonances with absolute certainty and high accuracy. The method can also tell when approximations fail to lie near true resonances, settling a decades-long controversy. Credit: Wikipedia / CC BY-SA 3.0

In the early '80s, several researchers were working to determine the location of atomic and molecular resonances, which are the frequencies at which atoms and molecules prefer to oscillate. Two groups of researchers (Rittby, et al., and Korsch, et al.), each using a different method, came up with different locations for these resonances. Settling the dispute proved to be extremely difficult due to the fact that neither method could predict the actual resonances, but instead simply gave approximations. In fact, at the time there was no way to locate resonances with absolute certainty.

The controversy remained unsettled for the last 30 years. But now in a new study, mathematicians have presented a that, for the first time, can locate resonances with absolute certainty and high accuracy. The method can also tell when approximations fail to lie near true resonances. Using this method, the researchers could show which approximations from the two groups in the '80s were accurate, and which were not, finally settling the controversy.

The mathematicians, Sabine Bögli, et al., from the University of Bern in Switzerland, Cardiff University in the UK, Stockholm University in Sweden, and Regensburg, Germany, have published their paper on the new method for determining resonances in a recent issue of the Proceedings of The Royal Society A.

A key ingredient in the new method is interval arithmetic. By operating on intervals rather than numbers, interval arithmetic allows every computational step to be carried out with absolute certainty. Interestingly, interval arithmetic has recently been used in a variety of areas, including navigation of an autonomous robotic sailboat, optimization of a spacecraft traveling between planets, and improvement of stability in particle accelerators.

In this study, the researchers used interval arithmetic to prove with absolute certainty that the approximations of resonances in Rittby, et al., do lie near true resonances, whereas the approximations of higher resonances in Korsch, et al., do not.

The new method also reveals additional information, in particular the existence of two new pairs of resonances that were originally suggested in a previous paper (Abramov, et al.), but not predicted by either Rittby, et al., or Korsch, et al. The researchers explained that this part was the most challenging because these pairs are so close to the imaginary axis, and the resonances in each pair are so close to each other. The results could have implications for a wide variety of areas.

"Resonances are ubiquitous in physics, e.g., in mechanics, acoustics, and electrical circuits," coauthor Marco Marletta, Professor at Cardiff University in the UK, told Phys.org. "In atomic physics, resonances are associated with metastable states of a system having a sufficiently long lifetime to be well-characterized and to render them significant experimentally and theoretically. The imaginary part of a is related to the decay rate or inverse lifetime of the system."

The researchers explained that the work could be applied to some very computationally challenging problems.

"For problems with complex eigenvalues (such as resonances in complex scaling), numerical computations are prone to be unreliable because small errors in input data may result in large errors in the results and/or because spurious eigenvalues may occur (spectral pollution)," explained coauthor Christiane Tretter, Professor at the University of Bern in Switzerland, and from Stockholm University in Sweden. "Therefore methods guaranteeing absolute certainty such as interval arithmetic may prove to be highly useful."

Explore further: Single-particle resonances in a deformed relativistic potential

More information: Sabine Bögli, et al. "Guaranteed resonance enclosures and exclosures for atoms and molecules." Proceedings of The Royal Society A. DOI: 10.1098/rspa.2014.0488

Related Stories

Quantum chaos in ultracold gas discovered

March 12, 2014

A team of University of Innsbruck researchers discovered that even simple systems, such as neutral atoms, can possess chaotic behavior, which can be revealed using the tools of quantum mechanics. The ground-breaking research, ...

Recommended for you

An inflexible diet led to the disappearance of the cave bear

August 23, 2016

Senckenberg scientists have studied the feeding habits of the extinct cave bear. Based on the isotope composition in the collagen of the bears' bones, they were able to show that the large mammals subsisted on a purely vegan ...

12 comments

Adjust slider to filter visible comments by rank

Display comments: newest first

Radley
3 / 5 (2) Nov 03, 2014
I don't claim to fully understand the content nor scope of this article so correct me if I am wrong, but does the Heisenberg-Gabor limit not state that what is claimed in this article to be impossible?
Torbjorn_Larsson_OM
5 / 5 (4) Nov 03, 2014
I'm not sure I understand. You refer to an uncertainty principle for the physics, but the resonances occur. And the analysis is on frequency alone.
Radley
5 / 5 (1) Nov 03, 2014
The Heisenberg-Gabor limit applies to harmonic analysis in general which states that you cannot achieve high temporal resolution and high frequency resolution at the same time. From how I understood it, it was a fundamental limit in signal processing. I think it's the beginning of the article that's a bit misleading since it says it can locate resonances with absolute certainty and high accuracy.
EyeNStein
5 / 5 (1) Nov 04, 2014
Heisenberg applies to uncertainty of individual 'particles'. These resonances apply to bulk systems, averaged over many particles. The more particles the more accurate the statistics.

I wonder if this interval technique could improve the predictions of instabilities in even larger bulk systems such as plasma confinement?
swordsman
not rated yet Nov 04, 2014
When the eigenvector values of the eigenvalues become variables themselves, analysis becomes much more difficult, although not impossible. Cut-and-try is not the full answer.
ralph638s
5 / 5 (2) Nov 04, 2014
Radley is right. The inverse relationship between time-resolution and frequency-resolution in signal processing is Heisenberg at macro-scale. The more you want to know about frequency, the less you can say about time, and vice-versa...
Da Schneib
5 / 5 (1) Nov 04, 2014
Ralph and radley have a point: this is, technically, the energy-time conjugation, under Heisenberg uncertainty. Frequency, of course, equates to energy by the Planck relation:
E = hf
where,
E is energy
h is Planck's Constant
and
f is frequency.

Thus, frequency determines energy, and energy and time are conjugate under uncertainty, so frequency and time are too.
Da Schneib
not rated yet Nov 04, 2014
Heisenberg applies to uncertainty of individual 'particles'. These resonances apply to bulk systems, averaged over many particles. The more particles the more accurate the statistics.
While this is true, it ignores the fact that the resonances, at base, must also apply to individual particles, or there would be no specific resonance to pick as "correct." In other words, they vary over the uncertainty from particle to particle, but not more than that; and by observing many particles one can come to a central, classical value. IOW, statistical behavior of this kind is classical, but the resonances are quantum.

I see some tension here; do you see it too?

I wonder if this interval technique could improve the predictions of instabilities in even larger bulk systems such as plasma confinement?
Now that is a great question.
ralph638s
1 / 5 (1) Nov 04, 2014
I remember reading an interesting paper on vixra by a guy with a Russian-sounding name from a university in Mexico who maintained that quantum mechanics was simply Fourier-transformed classical mechanics...
Da Schneib
5 / 5 (1) Nov 04, 2014
vixra
>wince<.
acronymous
5 / 5 (2) Nov 05, 2014
I don't claim to fully understand the content nor scope of this article so correct me if I am wrong, but does the Heisenberg-Gabor limit not state that what is claimed in this article to be impossible?

The certainty in question here is a matter of mathematics in the first instance, and a matter of physics secondarily, because the mathematics has physical implications.
It's like a question of determining which of two irrational numbers, each given by convergent series, is larger. If you round off while computing the respective series, the roundoffs may spoil your verdict. If you try to stay exact, you may run into an explosion in memory requirements that sinks you. With interval arithmetic, at each step you verify a fact of the form "such and so lies between A and B", where A and B are tractable sized rational numbers. A chain of such calculations can establish that the final answers lie one in [P,Q], the other in [R,S], with Q less than R. Win!
swordsman
not rated yet Nov 10, 2014
It is indeed true that the eigenvalues of the eigenvector equations show the exact nature of resonance and the precise Rydberg frequencies of radiation. It is an electromagnetic phenomenon that has been accurately described with proof. It depends on the model of the atom, and quantum mechanics doesn't work.

Please sign in to add a comment. Registration is free, and takes less than a minute. Read more

Click here to reset your password.
Sign in to get notified via email when new comments are made.