This article is devoted to the work of Paul Dirac, whose equation has significantly enriched quantum mechanics. It describes the basic concepts necessary to understand the physical meaning of an equation, as well as how to use it.
Science and scientists
A person who is not connected with science, represents the process of obtaining knowledge by some kind of magical action. And scientists, according to such people, are cranks who speak an incomprehensible language and are slightly arrogant. Acquiring a researcher, a person who is far from science immediately says that he did not understand physics at school. Thus, the layman fenced off from scientific knowledge and asks a more educated interlocutor to speak easier and more clearly. Surely Paul Dirac, whose equation we are considering, was welcomed in the same way.
Elementary particles
The structure of matter has always excited curious minds. In ancient Greece, people noticed that the marble steps, along which many legs passed, changed shape over time, and suggested: each foot or sandal carried away with it a tiny particle of matter. They decided to call these elements "atoms", that is, "indivisible." The name remained, but it turned out that both the atoms and the particles that make up the atoms are also compound, complex. These particles are called elementary. The work of Dirac is dedicated to them, the equation of which made it possible not only to explain the electron spin, but also to suggest the presence of an anti-electron.
Wave-particle duality
The development of photography technology in the late nineteenth century entailed not only a fashion for capturing oneself, food and cats, but also advanced the possibilities of science. Having received such a convenient tool as fast photography (recall, before exposure time reached 30-40 minutes), scientists began to massively record a variety of spectra.
Theories of the structure of substances existing at that time could not unambiguously explain or predict the spectra of complex molecules. At first, Rutherford's famous experiment proved that the atom is not so indivisible: in its center was a heavy positive nucleus, around which light negative electrons were located. Then the discovery of radioactivity proved that the nucleus is not a monolith, but consists of protons and neutrons. And then the almost simultaneous discovery of an energy quantum, the Heisenberg uncertainty principle and the probabilistic nature of the location of elementary particles gave impetus to the development of a fundamentally different scientific approach to the study of the world. A new section has appeared - particle physics.
The main question at the dawn of this century of great discoveries on an extremely small scale was the explanation of the presence of elementary particles and the mass and properties of the wave.
Einstein proved that even the elusive photon has mass, as it transmits momentum to the solid body it falls on (the phenomenon of light pressure). At the same time, numerous experiments on the scattering of electrons by slots indicated at least the presence of diffraction and interference in them; this is characteristic only of a wave. As a result, I had to admit: elementary particles are both an object with mass and a wave. That is, the mass of, say, an electron is “smeared” into a packet of energy with wave properties. This principle of wave-particle duality made it possible to explain first of all why the electron does not fall on the nucleus, and also for what reasons the orbits exist in the atom, and the transitions between them are spasmodic. These transitions give rise to a spectrum unique to any substance. Further, particle physics was supposed to explain the properties of the particles themselves, as well as their interaction.
Wave function and quantum numbers
Erwin Schrödinger made a surprising and still obscure discovery (Paul Dirac later built his theory on its basis). He proved that the state of any elementary particle, for example, an electron, is described by the wave function ψ. By itself, it does not mean anything, but its square will show the probability of finding an electron in a given place in space. In this case, the state of an elementary particle in an atom (or other system) is described by four quantum numbers. These are the main (n), orbital (l), magnetic (m) and spin (m s ) numbers. They show the properties of an elementary particle. An analogy is a bar of oil. Its characteristics are weight, size, color and fat content. However, the properties that describe elementary particles cannot be understood intuitively; they must be realized through a mathematical description. The work of Dirac, whose equation is the focus of this article, is devoted to the last, spin number.
Spin
Before proceeding directly to the equation, it is necessary to explain what the spin number m s means. It shows the intrinsic angular momentum of an electron and other elementary particles. This number is always positive and can take an integer value, zero or half-integer value (for an electron m s = 1/2). Spin is a vector quantity and the only one that describes the orientation of the electron. Quantum field theory lays the spin on the basis of exchange interaction, which has no counterpart in the usually intuitive mechanics. The spin number shows how the vector should rotate in order to return to its original state. An example is an ordinary ballpoint pen (let the writing part be the positive direction of the vector). For it to return to its original state, it must be rotated 360 degrees. This situation corresponds to a back equal to 1. With a 1/2 back, like an electron, the rotation should be 720 degrees. So, in addition to the mathematical instinct, one must have developed spatial thinking in order to understand this property. A little higher was the wave function. It is the main “actor” of the Schrödinger equation, with the help of which the state and position of an elementary particle are described. But this ratio in its original form is intended for particles without spin. It is possible to describe the state of an electron only if we generalize the Schrödinger equation, which was done in the work of Dirac.
Bosons and fermions
A fermion is a particle with a half-integer spin value. Fermions are located in systems (for example, atoms) according to the Pauli principle: in each state there should be no more than one particle. Thus, in an atom, each electron is somewhat different from all the others (some quantum number has a different meaning). Quantum field theory also describes another case - bosons. They have a whole spin and can all be simultaneously in one state. The realization of this case is called Bose condensation. Despite the rather well-confirmed theoretical possibility of obtaining it, in practice this was realized only in 1995.
Dirac equation
As we said above, Paul Dirac derived the equation of the classical field of an electron. It also describes the states of other fermions. The physical meaning of the relationship is complex and multifaceted, and many fundamental conclusions follow from its form. The form of the equation is as follows:
- (mc 2 α 0 + c ∑ a k p k {k = 0-3}) ψ (x, t) = i ħ {∂ ψ / ∂ t (x, t)},
where m is the mass of the fermion (in particular the electron), c is the speed of light, p k are the three momentum component operators (along the x, y, z axes), ħ is the truncated Planck constant, x and t are three spatial coordinates (correspond to the X axes , Y, Z) and time, respectively, and ψ (x, t) are the four-component complex wave function, and α k (k = 0, 1, 2, 3) are the Pauli matrices. The latter are linear operators that act on the wave function and its space. This formula is quite complicated. To understand at least its components, one must understand the basic definitions of quantum mechanics. You should also have remarkable mathematical knowledge to at least know what a vector, matrix and operator are. The form of the equation will tell the specialist even more than its components. A person knowledgeable in nuclear physics and familiar with quantum mechanics will understand the importance of this relationship. However, I must admit that the Dirac and Schrödinger equations are just the elementary foundations of a mathematical description of the processes that occur in the world of quantum quantities. Theoretical physicists who decided to devote themselves to elementary particles and their interaction should understand the essence of these relations in the first or second courses of the institute. But this science is fascinating, and it is in this area that you can make a breakthrough or perpetuate your name by assigning it to an equation, transformation or property.
The physical meaning of the equation
As we promised, we tell you what conclusions the Dirac equation holds for the electron. First, from this relation it becomes clear that the electron spin is ½. Secondly, according to the equation, an electron has its own magnetic moment. It is equal to the Bohr magneton (unit of elementary magnetic moment). But the most important result of obtaining this relation lies in the imperceptible operator α k . The derivation of the Dirac equation from the Schrödinger equation took a long time. At first, Dirac thought that these operators interfere with the relation. With the help of various mathematical tricks, he tried to exclude them from the equation, but he did not succeed. As a result, the Dirac equation for a free particle contains four operators α. Each of them is a matrix [4x4]. Two correspond to the positive mass of the electron, which proves the presence of two positions of its spin. The other two give a solution for the negative particle mass. The simplest knowledge in physics provides a person with the opportunity to conclude that this is impossible in reality. But as a result of the experiment, it turned out that the last two matrices are solutions for an existing particle, the opposite of an electron - an anti-electron. Like an electron, a positron (as this particle was called) has a mass, but its charge is positive.
Positron
As often happened in the era of quantum discoveries, Dirac at first did not believe his own conclusion. He did not dare to openly publish the prediction of a new particle. True, in many articles and at various symposiums, the scientist emphasized the possibility of its existence, although it did not postulate it. But soon after the conclusion of this famous ratio, the positron was found in cosmic radiation. Thus, its existence was empirically confirmed. Positron is the first element of antimatter found by people. A positron is born as one of the twins of a pair (the other twin is an electron) during the interaction of very high energy photons with the nuclei of matter in a strong electric field. We will not give figures (the interested reader will find all the necessary information himself). However, it is worth emphasizing that we are talking about cosmic scales. Only supernova explosions and collisions of galaxies can produce photons of the required energy . They are also found in some amounts in the nuclei of hot stars, including the Sun. But a person always strives for his own benefit. Annihilation of matter with antimatter gives a lot of energy. To curb this process and start it for the benefit of mankind (for example, engines of interstellar liners for annihilation would be effective), people learned how to make protons in laboratory conditions.
In particular, large accelerators (such as the hadron collider) can create electron-positron pairs. Earlier, it was also suggested that there are not only elementary antiparticles (besides an electron there are several more), but also a whole antimatter. Even a very small piece of any antimatter crystal would provide energy to the entire planet (maybe superman kryptonite was antimatter?).
But alas, the creation of antimatter is heavier than hydrogen nuclei in the observable universe has not been documented. However, if the reader thinks that the interaction of a substance (we emphasize precisely a substance, and not a single electron) with a positron immediately ends with annihilation, then he is mistaken. When a positron is braked at a high speed, a coupled electron-positron pair, called positronium, arises in non-zero probability in some liquids. This formation has some properties of an atom and is even able to enter into chemical reactions. But this fragile tandem does not exist for long and then all the same annihilates with the emission of two, and in some cases three gamma rays.
Disadvantages of the equation
Despite the fact that due to this ratio an antielectron and antimatter were discovered, it has a significant drawback. The equation record and the model built on its basis are not able to predict how particles are born and destroyed. This is a peculiar irony of the quantum world: a theory that predicted the birth of matter-antimatter pairs is not able to adequately describe this process. This flaw was eliminated in quantum field theory. By introducing the quantization of fields, this model describes their interaction, including the birth and destruction of elementary particles. By "quantum field theory" in this case we mean a completely specific term. This is a field of physics that studies the behavior of quantum fields.
Dirac equation in cylindrical coordinates
First, let us know what a cylindrical coordinate system is. Instead of the usual three mutually perpendicular axes, angle, radius and height are used to determine the exact location of a point in space. This is the same as the polar coordinate system on the plane, only the third dimension is added - the height. This system is convenient if you want to describe or explore some surface that is symmetrical about one of the axes. For quantum mechanics, this is a very useful and convenient tool that can significantly reduce the size of formulas and the number of calculations. This is a consequence of the axisymmetry of the electron cloud in the atom. The Dirac equation in cylindrical coordinates is solved somewhat differently than in the usual system, and sometimes gives unexpected results. For example, some applied problems to determine the behavior of elementary particles (most often electrons) in a quantized field were solved by converting the form of the equation to cylindrical coordinates.
Using the equation to determine the structure of particles
This equality describes simple particles: those that are not made up of even smaller elements. Modern science is capable of measuring magnetic moments with fairly high accuracy. Thus, the discrepancy between the value calculated using the Dirac equation and the experimentally measured magnetic moment will indirectly indicate a complex particle structure. Recall that this equality applies to fermions, their spin is half-integer. Using this equation, the complex structure of protons and neutrons was confirmed. Each of them consists of even smaller elements called quarks. A gluon field holds quarks together, preventing them from crumbling. There is a theory that quarks are not the most elementary particles of our world. But while people do not have enough technical power to check it.