Conclusions of Dirac. The Dirac equation. Quantum field theory

This article is devoted to the work of Paul Dirac, whose equation greatly enriched quantum mechanics. It describes the basic concepts necessary to understand the physical meaning of the equation, as well as the ways to use it.

Science and scientists

A person who is not connected with science, represents the process of extracting knowledge by some magical action. And scientists, in the opinion of such people, are freaks who speak an incomprehensible language and are slightly arrogant. Getting acquainted with the researcher, a person far from science immediately says that he did not understand physics at school. Thus the philistine fences himself off from scientific knowledge and asks a more educated interlocutor to speak easier and more clearly. For sure, Paul Dirac, whose equation we are considering, was welcomed the same way.

Elementary particles

The structure of matter has always stirred curious minds. In ancient Greece, people noticed that the marble steps, over which many legs had passed, changed shape over time, and suggested that each foot or sandal carried away a tiny particle of matter with it. These elements were decided to be called "atoms", that is, "indivisible." The name remains, but it turned out that both the atoms and the particles, of which the atoms are composed, are also compound, complex. These particles are called elementary particles. Dirac's work is devoted to them, the equation of which allowed not only to explain the electron spin, but also to assume the presence of an antielectron.

Corpuscular Wave Dualism

The development of photography in the late nineteenth century led not only to fashioning oneself, food and cats, but also promoted the possibilities of science. Having received such a convenient tool as a fast photo (recall, before the exposure lasted up to 30-40 minutes), scientists began to massively record a variety of spectra.

The theory of the structure of substances existing at that time could not unequivocally explain or predict the spectra of complex molecules. First, the famous experience of Rutherford proved that the atom is not so indivisible: in its center was a heavy positive nucleus, around which were located lightweight negative electrons. 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 the quantum of energy, 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 surrounding world. There was a new section - the physics of elementary particles.

The main issue at the dawn of this century of great discoveries in an extremely small scale was the explanation of the presence of elementary particles and mass and properties of the wave.

Einstein proved that even an elusive photon has mass, since it transmits an impulse to a solid body, which falls (the phenomenon of light pressure). At the same time, numerous experiments on the scattering of electrons on slits spoke at least about the presence of diffraction and interference in them, this is peculiar only to the wave. In the end, I had to admit: elementary particles are both an object with mass and a wave. That is, the mass of, say, the electron is "smeared" into a packet of energy with wave properties. This principle of corpuscular-wave dualism made it possible to explain first of all why the electron does not fall on the nucleus, and also for what reasons there are orbits in the atom, and the transitions between them are abrupt. These transitions generate a spectrum unique to any substance. Further, the physics of elementary particles should explain the properties of the particles themselves, as well as their interaction.

The wave function and quantum numbers

Erwin Schrödinger made a surprising and still incomprehensible discovery (on his foundation later, Paul Dirac built his theory). He proved that the state of any elementary particle, for example, of 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. The state of an elementary particle in an atom (or other system) is described by four quantum numbers. This is the main (n), orbital (l), magnetic (m) and spin (m _s ) numbers. They show the properties of an elementary particle. As an analogy, you can bring a bar of oil. Its characteristics - weight, size, color and fat. However, the properties describing elementary particles can not 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 is} . It shows the intrinsic moment of the 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 value and the only one that describes the orientation of an electron. Quantum field theory puts the spin at the basis of the exchange interaction, to which there is no analog in usually intuitive mechanics. The spin number shows how the vector should turn to come to the original state. An example is an ordinary ball-point pen (the writing part is a positive direction of the vector). To make it come to its original state, it must be rotated 360 degrees. This situation corresponds to a spin equal to 1. For a back of 1/2, like an electron, the rotation should be 720 degrees. So, besides the mathematical instinct, one must have developed spatial thinking in order to understand this property. A little bit above we talked about the wave function. It is the main "actor" of the Schrödinger equation, which describes the state and position of an elementary particle. But this relation in its original form is intended for particles without spin. We can describe the state of the electron only if we generalize the Schrodinger equation, which was done in Dirac's work.

Bosons and fermions

Fermion is a particle with a half-integral spin value. Fermions are located in systems (for example, atoms) according to the Pauli principle: in each state there should not be more than one particle. Thus, in an atom, each electron is somehow different from all the others (some quantum number has a different meaning). Quantum field theory describes another case - bosons. They have a whole spin and can all be in the same state at the same time. The realization of this case is called Bose condensation. Despite a fairly well-documented theoretical opportunity to obtain it, this was practically done only in 1995.

The 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 relation is complex and multifaceted, and many fundamental conclusions follow from its form. The form of the equation is:

- (mc ² α ₀ + c Σ 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 is the three operator of the momentum components (along the x, y, z axes), ħ is the truncated Planck constant, x and t are the three spatial coordinates (corresponding to the X , Y, Z) and time, respectively, and ψ (x, t) is the four-component complex wave function, α _k (k = 0, 1, 2, 3) - Pauli matrices. The latter are linear operators that act on the wave function and its space. This formula is quite complicated. To understand even its components, we must understand the basic definitions of quantum mechanics. Also, one should have remarkable mathematical knowledge in order to at least know what a vector, matrix and operator are. To the specialist, the form of the equation will say even more than its components. A person who is versed in nuclear physics and familiar with quantum mechanics will understand the importance of this relationship. However, we must admit that the Dirac and Schrödinger equations are merely elementary foundations of the mathematical description of 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 relationships in the first-second year of the institute. But this science is fascinating, and it is in this area that one can make a breakthrough or perpetuate his name, appropriating his equation, transformation or property.

The physical meaning of equation

As we promised, we tell what conclusions the Dirac equation for the electron has. First, from this relation it becomes clear that the electron spin is ½. Secondly, according to the equation, the electron has its own magnetic moment. It is equal to the Bohr magneton (unit of the 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 ratio. Using 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 existence of two positions of its spin. The other two give a solution for the negative mass of the particle. The simplest knowledge in physics gives a person the opportunity to conclude that this is impossible in reality. But as a result of the experiment it was found out that the last two matrices are solutions for the existing particle opposite to the electron - antielectron. Like an electron, a positron (as they called this particle) has a mass, but its charge is positive.

Positron

As was often the case in the era of quantum discovery, Dirac at first did not believe his own conclusion. He did not dare 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 he did not postulate it. But soon after the derivation of this famous relationship, the positron was found in cosmic radiation. Thus, its existence was confirmed empirically. The positron is the first element of antimatter found by humans. A positron is born as one of the twin pairs (the other twin is an electron) in the interaction of very high energy photons with matter nuclei in a strong electric field. We will not quote figures (the interested reader will find all the necessary information himself). However, it should be emphasized that we are talking about space scales. Only the explosions of supernovas and collisions of galaxies are able to produce photons of the necessary energy. They are also contained in a certain amount in the nuclei of hot stars, including the Sun. But people always strive for their own benefit. Annihilation of matter with antimatter gives a lot of energy. To curb this process and start it for the benefit of humanity (for example, the engines of interstellar liners on annihilation would be effective), people learned how to make protons in the laboratory.

In particular, large accelerators (such as the hadron collider) can create electron-positron pairs. Previously, it was also suggested that there are not only elementary antiparticles (there are several more besides the electron), but also an entire antimatter. Even a very small piece of any crystal from antimatter would provide energy to the entire planet (maybe superman kryptonite was antimatter?).

But alas, the creation of antimatter heavier than hydrogen nuclei in the foreseeable universe was not documented. However, if the reader thinks that the interaction of a substance (let us emphasize, it is the matter, and not the individual electron) with the positron ends immediately with annihilation, then it is mistaken. When the positron is decelerated at high speed, a bound electron-positron pair, called positronium, appears in certain liquids with nonzero probability. This formation has some properties of the atom and is even capable of entering into chemical reactions. But this fragile tandem does not last long and then still annihilates with the emission of two, and in some cases, three gamma quanta.

Disadvantages of the equation

Despite the fact that due to this relation an antielectron and antimatter were detected, it has a significant drawback. The record of the equation and the model constructed on its basis are not able to predict how particles are born and destroyed. This is a kind of irony of the quantum world: the theory that predicted the birth of matter-antimatter pairs is not capable of adequately describing this process. This deficiency was eliminated in quantum field theory. By introducing the quantization of fields, this model describes their interaction, including the creation and annihilation of elementary particles. By "quantum field theory" in this case is meant a completely specific term. This is a field of physics that studies the behavior of quantum fields.

The Dirac equation in cylindrical coordinates

To begin with, let's say what a cylindrical coordinate system is. Instead of the usual three mutually perpendicular axes, the 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 - the height is added. This system is convenient if it is required to describe or investigate a certain surface symmetrical with respect to one of the axes. For quantum mechanics, this is a very useful and convenient tool that allows you to 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 customary system, and sometimes gives unexpected results. For example, some applied problems for determining the behavior of elementary particles (most often electrons) in a quantized field were solved by transforming the form of the equation to cylindrical coordinates.

The use of the equation for determining the structure of particles

This equation describes simple particles: those that do not consist of even smaller elements. Modern science is able to measure magnetic moments with a sufficiently high accuracy. Thus, the discrepancy between the value calculated by the Dirac equation and the experimentally measured magnetic moment will indirectly indicate the complex structure of the particle. We recall that this equation is applicable to fermions, their spin is half-integral. Using this equation, the complex structure of protons and neutrons was confirmed. Each of them consists of even smaller elements, which are called quarks. The gluon field keeps the quarks together, not allowing them to crumble. 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.