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Science News

Quantum Mechanics by Computer

Dietrick E. Thomsen

Published Oct 23, 1982• 2 citations
Publisher

Abstract

Wave equations -wave functions -are the mathematical way of describing the behavior of things in quantum mechanics. The physical basis of this is that on the microscopic level every object is at the same time both a particle a bundle of waves. So the mathematics that describes dynamics on this level turns out to be the mathematics of wave functions. Wave functions are far from being the easiest things in mathematics to solve compute with. To calculate the behavior of an atomic nucleus, for instance, involves working with dozens or hundreds of wave functions (each proton neutron has its own) that are related to one another in such a way that the solution to each one depends on all the others. They cannot be solved one by one; they have to be solved ensemble, dozens or hundreds of simultaneous equations, as mathematicians would describe By hand the task is formidable, perhaps practically impossible. However, at the Lawrence Livermore National Laboratory, a Cray 1, one of the latest, biggest fastest of computers has now come to the rescue. Using it M. S. Weiss co-workers have been able to calculate what happens in various kinds of collisions of a krypton nucleus with a lanthanum nucleus. Then, with the latest computer graphics they are able to display these processes in an animated film. The two nuclei approach each other, collide, in some cases fuse, in others come apart, everything is decorated with bright false colors representing the density levels of matter inside the nuclei. The pictures the information in them represent the solution of 146 simultaneous wave functions, one for every two nucleons, every two particles in the two nuclei. There is one function for every two nucleons rather than every one because the mathematics contains one important approximation. Nucleons can come with two directions of spin, spin up or spin down. This theory does not distinguish between the two spin states, so each function actually represents the average of one of each. The pictures show a dozen instances, starting from a head-on collision proceeding to instances of greater greater offset between the centers of the two nuclei until they arrive at a sideswipe in which the nuclei hardly touch each other. The collisions near head-on result in fusion. A little offset contributes an angular momentum: The fusion product rotates. More offset the nuclei collide, losing a lot of energy, but retaining enough to come apart again. This is known as deep inelastic scattering. Finally, an instance where the two nuclei do not touch, but where distortions vibrations are produced in them by the force between them when they are close. Weiss says the film wakes physicists up when he shows it at the end of a talk. He mentions one experimentalist who came up said how nice it was to be able to actually see these things. At least four levels of abstraction, Weiss comments, and he talks about seeing it. Yet viewing the film, one can understand what the experimentalist meant. The calculations behind the film are quite accurate in predicting the main events of the collisions, that is what is really important. According to Weiss, they predict accurately the conditions under which fusion is likely, the proper range of deep inelastic scattering so forth. In the pictures the behavior looks like that of a fluid, liquid drops coming together. Weiss stresses that it is actually the sum of discrete motions, those 146 individual wave functions. One of the purposes of the exercise was to see whether it was possible to derive a macroscopic theory that is, one that would give a simpler mathematical expression for the over-all behavior of the nuclei, that would yield the desired predictions without having to start from the 146 individual wave functions. It turns out, Weiss says, that a macroscopic theory is possible in some particular cases, but the microscopic one, starting from the individual wave functions, remains the general one. Computation was the serious hindrance to doing it before now. As Weiss says, the theory itself was written down years ago by P. A. M. Dirac, one of the pioneer theorists of quantum mechanics. It took Weiss co-workers a year to figure out how to feed the computation to the Each of the dozen collisions which takes about 10-21 seconds in real time, but is displayed in extreme slow motion in the film-took an hour of computation time using the core of the Cray That's expensive computing, but, says Weiss, I would never try it on a lesser computer. O

DOI
10.2307/3966992

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Wave equations -wave functions -are the mathematical way of describing the behavior of things in quantum mechanics. The physical basis of this is that on the microscopic level every object is at the same time both a particle a bundle of waves. So the mathematics that describes dynamics on this level turns out to be the mathematics of wave functions. Wave functions are far from being the easiest things in mathematics to solve compute with. To calculate the behavior of an atomic nucleus, for instance, involves working with dozens or hundreds of wave functions (each proton neutron has its own) that are related to one another in such a way that the solution to each one depends on all the others. They cannot be solved one by one; they have to be solved ensemble, dozens or hundreds of simultaneous equations, as mathematicians would describe By hand the task is formidable, perhaps practically impossible. However, at the Lawrence Livermore National Laboratory, a Cray 1, one of the latest, biggest fastest of computers has now come to the rescue. Using it M. S. Weiss co-workers have been able to calculate what happens in various kinds of collisions of a krypton nucleus with a lanthanum nucleus. Then, with the latest computer graphics they are able to display these processes in an animated film. The two nuclei approach each other, collide, in some cases fuse, in others come apart, everything is decorated with bright false colors representing the density levels of matter inside the nuclei. The pictures the information in them represent the solution of 146 simultaneous wave functions, one for every two nucleons, every two particles in the two nuclei. There is one function for every two nucleons rather than every one because the mathematics contains one important approximation. Nucleons can come with two directions of spin, spin up or spin down. This theory does not distinguish between the two spin states, so each function actually represents the average of one of each. The pictures show a dozen instances, starting from a head-on collision proceeding to instances of greater greater offset between the centers of the two nuclei until they arrive at a sideswipe in which the nuclei hardly touch each other. The collisions near head-on result in fusion. A little offset contributes an angular momentum: The fusion product rotates. More offset the nuclei collide, losing a lot of energy, but retaining enough to come apart again. This is known as deep inelastic scattering. Finally, an instance where the two nuclei do not touch, but where distortions vibrations are produced in them by the force between them when they are close. Weiss says the film wakes physicists up when he shows it at the end of a talk. He mentions one experimentalist who came up said how nice it was to be able to actually see these things. At least four levels of abstraction, Weiss comments, and he talks about seeing it. Yet viewing the film, one can understand what the experimentalist meant. The calculations behind the film are quite accurate in predicting the main events of the collisions, that is what is really important. According to Weiss, they predict accurately the conditions under which fusion is likely, the proper range of deep inelastic scattering so forth. In the pictures the behavior looks like that of a fluid, liquid drops coming together. Weiss stresses that it is actually the sum of discrete motions, those 146 individual wave functions. One of the purposes of the exercise was to see whether it was possible to derive a macroscopic theory that is, one that would give a simpler mathematical expression for the over-all behavior of the nuclei, that would yield the desired predictions without having to start from the 146 individual wave functions. It turns out, Weiss says, that a macroscopic theory is possible in some particular cases, but the microscopic one, starting from the individual wave functions, remains the general one. Computation was the serious hindrance to doing it before now. As Weiss says, the theory itself was written down years ago by P. A. M. Dirac, one of the pioneer theorists of quantum mechanics. It took Weiss co-workers a year to figure out how to feed the computation to the Each of the dozen collisions which takes about 10-21 seconds in real time, but is displayed in extreme slow motion in the film-took an hour of computation time using the core of the Cray That's expensive computing, but, says Weiss, I would never try it on a lesser computer. O
The process of observation disturbs the thing being observed and that disturbance might possibly affect other observations. Heisenberg showed that simultaneous measurements of an electron's position and its momentum interfere with each other and thus induce essential uncertainties into those measurements. Quantum mechanics represents the behavior of electrons around nuclei in terms of mathematical constructs called wave functions, or orbitals, which contain all possible information about the electron. The square of the wave function gives the probability of finding the electron in a given region of space. For mathematical reasons, physicists and chemists almost always use the abstract concept of wavefunctions (orbitals), rather than that of electron probability clouds, to describe electrons. The distribution of probability within a probability cloud is not uniform. For some, the greatest probability is near, or even at, the nucleus
A picture is worth a thousand words" is one of the rare maxims phrased as an understatement.A high-resolution colour picture contains in fact more information (bit-wise) than a million words.It is in gene ral also faster, easier and more pleasant to grasp than mere words.Our brain is equipped with a rapid and efficient mechanism dedicated to analysing pictures and extracting meaning from them.In spite of our tremendous handicap in arith metic computations, we can still easily beat any computer when it comes to versatile image recognition.We are indeed so good at it that we like the exercise even better when the images are projected at the rate of at least 24 per second, in the form of video or film.For better or for worse, the dominant means of communication today is the ani mated image, and it is here to stay.But there are things we, as humans, would rather express in words, and there are things -quantum mechanics and field theory among them -that we, as scientists, would express more universally in equations and numbers.This should come as no sur prise, you might say: any fundamental theo ry is naturally presented in terms of defini tions, equations and constants of nature.True: but any fundamental theory is then developed and taught with the help of examples, demonstrations, experiments, applications, and images.Practitioners of quantum mechanics somehow managed with few of the last.The reasons for the scarcity of quantum images have very different origins, some valid, some unjustified.The following re marks illustrate the situation: -Quantum mechanics is manifest mainly at the molecular, atomic and subatomic scales.Only recently did microscopes beco me capable of producing "real" images at these scales.Some isolated coherent quan tum effects are known at larger scales (neu tron stars, lasers, superconductivity, liquid helium, etc.), but real images demonstrating clearly their quantum aspect are lacking.-In the domain of "theoretical" images, an overwhelming hurdle has been the rather
Much has been talked about quantum computers. Today they are working on difficult simulations, so we can already congratulate the college of researchers at the Oak Ridge National Laboratory as the world's first to successfully simulate an atomic nucleus using a quantum computer. The results that were published in the physical examination letters demonstrate the huge capacity of quantum systems to calculate the problems of nuclear physics and to be able to be successfully used in the future for special and complex simulations. It is already known that quantum calculus, which is based on the quantum principles of matter and which was proposed by American physicist Richard Feynman in the early 1980s, unlike the ordinary bits of normal computers today, uses the qubit units used by quantum computers to store information in two-state systems, such as electrons or photons, considered simultaneously in all possible quantum states (essentially a phenomenon known as overlapping). In October 2017, a multidisciplinary ORNL team began working on codes to perform various simulations on quantum computers such as IBM QX5 and Righetti 19Q within a DOE Quantum Testbed Pathfinder project in an effort to verify and validate various scientific applications on different types of quantum hardware. Today, quantum computers have potential applications in cryptography, artificial intelligence and weather forecasting, as each additional qubit clings in some way inextricably from the others, thus exponentially increasing the number of possible results for the final measured state. Obviously, this huge benefit also has negative effects on the system, because errors can in turn exponentially increase the size of the problem and the final results even. However, it is hoped that in the future on the basis of much-improved hardware, researchers will be able to solve problems that cannot be solved today using current calculation methods. In the present paper the authors propose an original method of rapid theoretical determination with great precision of the dimensions of all elementary particles, the method which, although predicting results similar to those obtained by other theoreticians, still has the advantage of its simplicity, of its general applicability, that is of its universality and especially this new method is a precious tool in the work of scientists worldwide to understand matter at its elementary level. Even though we would have expected that the size of a deuteron would be larger (double) than that of one component particle (of a proton or neutron), at the same velocity, the theory (20-21) demonstrates the opposite. By fusing two free nucleons, a proton and a neutron, a deuterium nucleus is obtained with smaller dimensions than its two components, which makes the energy of rejection between two Deuteron cores much larger than we would have waited. That's why the fusion reaction between two Deuterons is even more difficult than was suspected, even when they are accelerating.
Classical mechanics is usually valid at the macro scale. The changes in the position and the momentum of the object over time are described by the Hamiltonian equation of motion. The wave function in quantum mechanics, if unobserved, evolves in a smooth and continuous way according to the Schrodinger equation, which is related to the Hamiltonian equation of motion. This equation describes a linear superposition of different states at time. A quantum mechanical description of a physical system is related to a probabilistic representation; it is described by a vector in Hilbert space. This description extends the two- or three-dimensional Euclidean space into spaces that have any finite or infinite number of dimensions. The state of a compound system is projected to the subspace that corresponds to the observed state and the vector representing the state is renormalized to the unit length. An observable describes a subspace for some dimensions with a special case of one dimension.
The Quantum theory deals with the microscopic structure of the Universe that describes the finer details of nature at the subatomic domain.To a physicist, quantum mechanics is one of the three great pillars supporting our intuition of evocative, bewildering and fascinating natural World as we continuously struggle with the inescapable strangeness of the subatomic domain as if something strange is always afoot.It is well accepted fact that quantum mechanics is a physical theory that describes the way things behave at subatomic level which is quite dazzling in its precision and explanatory power.A test of quantum electrodynamics, the oldest and best understood of the modern quantum theory that involves measuring the way an electron behaves in the vicinity of a magnetic force. Many of the technological and social changes that have revolutionized our lives have actually risen out of the fundamental theoretical research carried out with only motivation to understand the World in a better way than ever. For example, the invention of transistor actually came with the fundamental theoretical research on the Uncertainty Principle which states that the uncertainty of the position of a particle times the uncertainty of its velocity is always a constant. If we double the certainty of the position of the particle, we have to make half the certainty of its velocity and vice versa. Quantum theory is perhaps the prime example of the infinitely esoteric in the sense that it describes a World in which a particle really can be in several places at once and it moves from one place to another by exploring the entire Universe simultaneously-the theory of simultaneity. The diverse and complex Universe is actually made of a few tiny particles according to the rules of quantum theory which is so simple that can be summarized on the back of an envelope. The more we understand about the elemental nature of the Universe, the simpler it looks. We have already discovered that everything in the diverse and complex Universe is nothing but the different assemblies of atoms, and the wide variety of atoms are constructed using the three building blocks at the sub-atomic domain i.e. subatomic particles: protons, neutrons and electrons. We have also discovered that protons and neutrons ate made up of combination of three entities called quarks. And that is where things stop as far as recent discovery is concerned. But the recent progress in particle physics inspiring the quest for the ultimate unified theory of physics that can describe everything in the Universe in one go. The discovery of Boson particle certainty will revolutionaries the Newton's mass concept and gravity into whole new looks. To an optimistic point of view, we must hope the best out of the progress in physics.
Quantum mechanics is a theory to explain and predict the behavior of particles such as electrons, protons, neutrons, atomic nuclei, atoms, and molecules, as well as the photon–the particle associated with electromagnetic radiation or light. From quantum theory we obtain the laws of chemistry as well as explanations for the properties of materials, such as crystals, semiconductors, superconductors, and superfluids. Applications of quantum behavior give us transistors, computer chips, lasers, and masers. The relatively new field of molecular biology, which leads to our better understanding of biological structures and life processes, derives from quantum considerations. Thus, quantum behavior encompasses a large fraction of modern science and technology. Quantum theory was developed during the first half of the twentieth century through the efforts of many scientists. In 1926, E. Schrödinger interjected wave mechanics into the array of ideas, equations, explanations, and theories that were prevalent at the time to explain the growing accumulation of observations of quantum phenomena. His theory introduced the wave function and the differential wave equation that it obeys. Schrödinger's wave mechanics is now the backbone of our current conceptional understanding and our mathematical procedures for the study of quantum phenomena. Our presentation of the basic principles of quantum mechanics is contained in the first three chapters. Chapter 1 begins with a treatment of plane waves and wave packets, which serves as background material for the subsequent discussion of the wave function for a free particle.