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Physics

The force of vacuum

Team from USP discovers the limits to the use of the Schrödinger wave equation, one of those most employed in the study of the atom

NASAMagnetic field on a cylinder: similar to a vacuumNASA

Son of a physician, Aristotle learnt at an early age to show assurance in all that he said. Later on, the Greek sage formulated the thesis that nature and a vacuum do not match. Almost 3,000 years later, though, vacuum has ceased to be a synonym for emptiness and has become an inexhaustible reservoir of energy that can no longer be disregarded. It is the vacuum that supplies energy to the electrons – atomic particles with a negative electrical charge – and keeps them in movement around the atomic nucleus. Aristotle would be surprised if he knew that, without this energy, objects would not have formed. There would be nothing besides a soup of electrons and protons – particles with a positive charge – that would not successfully organize themselves into atoms.

It was by taking the vacuum into account that researchers from the Physics Institute of the University of São Paulo (USP) compared the result for the minimum energy of the atom calculated using the equations formulated by Heisenberg and by Schrödinger. Both equations were proposed at the beginning of last century, when vacuum was still seen as an empty space, by two exponents of modern science, the German Werner Heisenberg and the Austrian Erwin Schrödinger. The physicistsfrom USP found that only Heisenberg’s formulation works in a satisfactory way, when the forces of vacuum are considered.

Accordingly, they established the limits for using one of the formulas most used in the study of the behavior of atoms, calledthe Schrödinger wave equation – including vacuum, Schrödinger’s equation produces an incorrect value for the minimumenergy of the atom. At the same time, the physicists from São Paulo did away with the ancient idea that the two approaches would always be equivalent, and would lead to the same result about the minimum energy of the atom – essential information for understanding, for example, at what temperature a metal melts.

Students, engineers and physicists are certainly going to like this news, because the equation that indicates correctly the energy of the atom is Heisenberg’s, simpler and easier to work with than the other. Schrödinger’s wave equation continues to be useful, but dealing with it will call for a bit more attention from now on. “In a way that is still not understood, the effects of the forces of vacuum are somehow included in some element of Schrödinger’s formulation”, observes Coraci Pereira Malta, one of the authors of the study, published in December in Physics Letters A.

“We just do not know how Schrödinger managed it, as he didn’t know the forces of the vacuum.” The work – inco-authorship with Humberto França and two of his students, Alencar Faria and Rodrigo Sponchiado, both from USP – demonstrates that the result of Heisenberg’s equation including the vacuum is equivalent to Schrödinger’s without the vacuum. “We proved that Schrödinger was really wrong, when he said that his formulation and Heisenberg’s wereequivalent”, Coraci comments. So as not to appear pretentious to the point of dethroning one of the founders of quantum mechanics, she drew up a more modest version to explain what they had done: “We confirmed what Dirac had suspected.”

Solitary particles
Briton Paul Dirac rediscovered vacuum in 1927. 20 years had already passed by since this kind of energy had been proposed by Germany’s Max Planck, the discoverer of the strange world of quantum mechanics, in which atomic particles acquire apparently absurd behaviors – being able to be in two places at the same time, or going from one point to another without passing through the middle. In 1963, Dirac raised the possibility that the two routes for calculating the minimum energy of the atom, Heisenberg’s and Schrödinger’s formulations, would not lead to the same result. Dirac arrived at this conclusion availing himself of sophisticated calculations, which described the interaction of the electron with the electromagnetic forces of the vacuum. His conclusions remained unknown until they were redeemed by the group from USP last year.

The São Paulo physicists arrived at the same result as Dirac with a very simple conceptual model, equivalent to an electron immersed in a vacuum, vibrating on the tip of a spring. It is the so-called simple harmonic oscillator with an electric charge, the same that can represent the hydrogen atom, with only one electron orbiting around a solitary particle – a proton – that constitutes the nucleus. This model also represents the atoms of another six chemical elements – lithium, sodium, potassium, rubidium, cesium and francium – in whose outermost layer there is just one electron.

Based on this test platform, the researchers assessed, first, the equation created by Heisenberg. Oddly enough, the mathematical route proposed by the German scientist to find the energy of the electron is similar to the approach adopted 300 years before by British physicist Isaac Newton to define the movement of any body. Heisenberg and Newton, each in his own way, see acceleration as an effect of force acting on this body and therefore foresee movement, whether of a component of an atomic nucleus, or of a planet. Newton, in 1687, and Heisenberg, in 1925, consider the position of the particle evolving in time – a variable abolished in Schrödinger’s equation, formulated one year later, in 1926.

Schrödinger works with states of movement independent of time – the electron is no longer seen as a particle, but as a wave. His focus resulted in an equation that is rather more complicated than Heisenberg’s. But Schrödinger guaranteed: both would lead to the same results. It was not, though, what was ascertained at the end of last year. The physicists from USP added the forces of vacuum to Heisenberg’s formula, and without any problem arrived at the correct value for the minimum energy of the electron oscillating on the tip of a spring. Doing the same with Schrödinger’s formulation , they noted that the energy of the electron simply doubled and led to strange situations – it would be like saying that an ordinary person almost 2 meters tall can be up to 4 meters in height. Schrödinger’s equation only worked without the forces of vacuum – the maximum height of the people went back to be 2 meters. That is where the practical recommendation comes from: do not add the forces of vacuum to the Schrödinger wave equation.

Except in this situation, vacuum can no longer be disregarded. The electron itself seems to perceive this kind of energy dispersed in space, like a fly soaking itself with humidity before the rain starts to fall. “The electron emits and absorbs radiation from the vacuum all the time”, França says. “And it maintains the orbit stable because it emits the same amount ofenergy as it absorbs.” Although almost imperceptible at room temperature, vacuum acts in a way that is similar to a magnetic field, resulting from the action of a common magnet, and manages to draw together two neutral metallic plates, in parallel and kept at a temperature close to absolute zero (-273o Celsius), as has already been demonstrated experimentally – it is the Casimir force, identified in 1954.

Even if it is not known as well as electricity, vacuum is more intense than gravity, the most tenuous and all-embracing of the forces that govern the Universe. The Casimir force becomes 16 times greater if the distance between the two plates falls byhalf, while gravity only multiplies fourfold. It still remains to be proved, but one imagines that vacuum can be the mysterious dark energy, which corresponds to 73% of the universe. Straight away, besides explaining the composition of the cosmos, vacuum has become important for representing a usable form of energy, even to replace electricity. This possibility arose only a few years after the quantum theory had become consistent, as a result of the joint work of a group of notable scientists that included Dirac, Schrödinger and Heisenberg. In 1928, when the phenomena that were to allow the construction of sound and television apparatuses were beginning to be clarified, American Harold Nyquist foresaw that vacuum could interfere with electric circuits.

On the basis of this idea, França, from USP, and a team from two American companies, Mission Research and ManyOne Networks, designed a piece of equipment that, if it works out, will successfully extract useful energy from a vacuum. The apparatus consists of a coil with a diameter of 2 centimeters, cooled down to -270o Celsius, which ought to work like an antenna to capture energy from the vacuum. “Depending on the way it is wound, the coil, in principle, can cancel out or increase the power for capturing energy from a vacuum”, says França. The plan is to set up and test the experiment before the year is out, provided that the problems with the budget are overcome. The prototype should not work out for less than R$ 150,000.

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