Insights Of Dirac. The Dirac Equation. Quantum field theory

This article focuses on the work of Paul Dirac, the equation of which has greatly enriched quantum mechanics. It describes the basic concepts necessary to understand the physical meaning of the equations, and the methods of its application.

Science and scientists

Man, not associated with science, is the process of extraction of knowledge in some magical action. And scientists, according to these people, is the eccentrics who speak in a strange language and a little arrogant. Getting acquainted with the researcher, far from science man immediately says he is in school physics is not understood. Thus the average person is fenced off from scientific knowledge and asks more educated interlocutor to speak more simply and clearly. Certainly Paul Dirac, the equation we are considering, welcomed as well.

Elementary particles

The Structure of the substance always worried about inquisitive minds. In Ancient Greece, people noticed that the marble stairs, which passed many feet, over time, change shape, and suggested: each foot or sandal carried a tiny particle of matter. These elements decided to call “atoms”, that is ‘indivisible". The name remained, but it became clear that the atoms and particles that make up atoms - are also composite, complex. These particles are called elementary. Focus on the work of Dirac equation which allowed not only to explain the electron spin, but also to assume the existence of the antielectron.

Wave-Particle duality

The Development of photographic techniques in the late nineteenth century entailed not only the fashion for the seal itself, food and cats, but also advanced science. Having such a convenient tool like quick photo (we will remind, earlier exposure reached up to 30-40 minutes), scientists began to capture a variety of spectra.

Recommended

"Knowledge is light and ignorance is darkness": the value, meaning and alternatives

There are some sayings that would seem to need no explanation, such as “teaching & ndash; light and ignorance – darkness”. But some still do not understand their meaning. But not only for such people is written by our article. I...

What was invented by Mendeleev for the army. The history and fate of the invention

D. I. Mendeleev was a brilliant Russian scientist-polymath, who made many important discoveries in various fields of science and technology. Many people know that he is the author of “Fundamentals of chemistry" and the periodic law of chem...

The origin of the Slavs. The influence of different cultures

Slavs (under this name), according to some researchers, appeared in the story only in 6 century ad. However, the language of nationality bears the archaic features of the Indo-European community. This, in turn, suggests that the origin of the Slavs h...

The Existing theory of the structure of the substances could not clearly explain or predict spectra of complex molecules. First, the famous experiment of Rutherford proved that the atom was not so indivisible: in its center was a heavy positive nucleus, around which a light negative electrons. Then the discovery of radioactivity proved that the kernel is not a monolith, but consists of protons and neutrons. And then the almost simultaneous discovery of the energy quantum, Heisenberg's 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 is a new section – physics of elementary particles.

A Major issue at the dawn of this century of great discoveries in ultra-small scales to explain the existence of elementary particles and masses, and properties of waves.

Einstein proved that even the elusive photon has a mass, as the momentum of a solid body, which is falling (the phenomenon of light pressure). While numerous experiments on the scattering of electrons by said slits at least about the existence of diffraction and interference, is characteristic only of waves. In the end had to admit that elementary particles simultaneously and object with mass, and wave. What is the mass of, say, an electron matter how “smeared” a package of energy with wave-like properties. This principle of wave-particle duality helped to explain primarily why the electron does not fall into the nucleus, as well as the reasons why in the atom there are orbits, and the transitions between them are abrupt. These transitions and give rise to a spectrum that is unique for any substance. Further particle physics was to explain the properties of the particles themselves and their interaction.

The Wave function and quantum numbers

Erwin schrödinger made a surprising and still poorly understood (on the basis of his later Paul Dirac constructed his theory). He proved that the state of any elementary particle, for example, describes an electron wave function ψ. In itself it means nothing, but its square will show the probability of finding the electron at a given place in space. The status of elementary particles in the atom (or other system) is described by four quantum numbers. This is the main (n), orbital (l), magnetic (m) and spin (m_S) number. They show the properties of elementary particles. As an analogy, allow the bar oil. Its characteristics-weight, size, color and oiliness. However, the properties that describe elementary particles cannot be understood intuitively, they need to realize using a mathematical description. The work of the Dirac equation which is the focus of this article on the last spin number.

Spin

Before proceeding to the equation, it is necessary to explain, what does the spin number m_S. It shows the angular momentum of the electron and other elementary particles. This number is always positive and can take an integer value, zero or palusalue value (for electron m_S = 1/2). Spin – the magnitude of the vector and the only one that describes the orientation of the electron. Quantum field theory puts the spin in the exchange interaction, which has no analogue in the usual intuitive mechanics. Spin number shows howway to turn vector to come to its original condition. An example is a ball pen (the writing part let it be the positive direction of the vector). To come to its original state, it should rotate 360 degrees. This situation corresponds to a spin equal to 1. When the spin 1/2 as the electron, the rotation has to be 720 degrees. Thus, besides mathematical intuition, it is necessary to have developed spatial thinking to understand this property. Slightly above discussed wave functions. She is the main “protagonist” of the schrödinger equation, which describes the condition and position of elementary particles. But this ratio in its original form is designed for particles without spin. To describe the state of the electron is possible only if the generalization of the schrödinger equation that has been done in the work of Dirac.

Bosons and fermions

Fermion-a particle with half-integer value of spin. The fermions are arranged in systems (e.g. atoms) according to the Pauli principle: each state must be no 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 spin and can all simultaneously be in the same state. The implementation of this case is called Bose-condensation. Despite the fairly well-confirmed theoretical possibility to obtain, in practice it is implemented only in 1995.

Dirac Equation

As we said above, Paul Dirac derived the equation for the classical field of the electron. It also describes the status of other fermions. The physical meaning of the relation is complex and multifaceted, and its shape should many fundamental insights. The equation is as follows:

- (mc² α₀+c ∑ a_KP_K {k=0-3}) ψ(x, t)= i ħ{ ∂ ψ/∂ t (x, t)},

Where M — the mass of the fermion (e.g. electron), With — the speed of light, p_K— the first component of the pulse (axes x, y, z), ħ — stripped down Planck's constant, x and t – three spatial coordinates (correspond to the axes X, Y, Z) and time, respectively, and ψ(x, t) — four-component complex wave function, α_K (k=0, 1, 2, 3) - the matrix Pauli. 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, it is necessary to understand the basic definitions of quantum mechanics. You should also have a remarkable mathematical knowledge to at least know what a vector, a matrix and an operator. A specialist form of the equation will tell even more than his components. People well versed in nuclear physics and is familiar with quantum mechanics will understand the importance of this ratio. However, I must admit that of the Dirac equation and schrödinger is just the basic Foundation of the mathematical description of the processes that occur in the world of quantum variables. Theoretical physicists, who decided to devote himself to the elementary particles and their interactions, must understand these ratios in the first and second courses of the Institute. But the science is fascinating, and in this area it is possible to make a breakthrough or to immortalize his name by setting the equation, converting or property.

Physical meaning of the equation

As we promised, tell, what holds the Dirac equation for the electron. First, from this relation it becomes clear that the spin of the electron is equal to ½. Secondly, according to the equation, electrons have an intrinsic magnetic moment. It is equal to the Bohr magneton (unit of elementary magnetic moment). But the main result of this ratio lies in the invisible operator α_K. The output of the Dirac equation from Schrodinger's equation took a lot of time. Initially, Dirac thought that these operators interfere with the ratio. Using different mathematical tricks he was trying to exclude them from the equation, but he failed. In the end, the Dirac equation for a free particle contains four operator α. Each of them is a matrix [4x4]. Two correspond to the positive mass of the electron, which proves the presence of two provisions in his back. The other two give the solution for negative mass particles. The simplest knowledge of physics give the man a chance to conclude that it is impossible in reality. But in the experiment it turned out that the last two matrix are the solutions to the existing particles, opposite to the electron – the antielectron. Like an electron, a positron (the so-called the particle) has a mass but its charge is positive.

Center

As often happened in the era of quantum discoveries, Dirac did not at first believe his own conclusion. He did not dare openly to publish a prediction of a new particle. However, in many articles and at various symposia scholar emphasized the possibility of its existence, although not postulated it. But soon after the withdrawal of this famous ratio, the positron was found in cosmic radiation. Thus, its existence has been confirmed empirically. Center – first found by the people of the element of antimatter. Center is born as one of a pair of twins (other twin – an electron) in the interaction with photons of very high energy nuclei of matter in a strong electric field. To bring the numbers, we will not (the interested reader and he will find all the necessary information). However, it should be emphasized,what we are talking about cosmic scale. To produce photons of desired energy are capable only of supernova explosions and collisions of galaxies. Also, they are in a number are contained in the cores of hot stars, including the Sun. But man is always striving for their own benefit. The annihilation of matter with antimatter gives a lot of energy. To curb this process and to use it for the benefit of humanity (for example, effective would be the engines of interstellar ships annihilation), people learned to make protons in the laboratory.

In particular, large accelerators (like the LHC) can create a pair electron-positron. Earlier also speculated that there are not only elementary antiparticle (electron in addition to their a few others), but an antimatter. Even a very small piece of any crystal of antimatter would provide energy to the entire planet (maybe kryptonite Superman was the anti-matter?).

But alas, the creation of antimatter nuclei heavier than hydrogen in the observable universe undocumented. However, if the reader thinks that the interaction of the substance (note, it is substances, not a single electron) positron annihilation immediately ends, he is mistaken. When braking the center with high velocity in some liquids with non-zero probability of occurring coupled pair electron-positron, called positronium. This education has some properties of an atom and is even able to enter into chemical reactions. But there is this delicate tandem for a short time and then still annihilates with emission of two, and in some cases three gamma rays.

Disadvantages of equations

Despite the fact that the ratio was discovered the antielectron antimatter, it has a significant drawback. Record the equations and the model constructed on its basis, are not able to predict how born and destroyed particles. It is a peculiar irony of the quantum world: the theory that predicted the birth of pairs of matter-antimatter, not able to adequately describe this process. This disadvantage was eliminated in the quantum field theory. By introducing quantized fields this model describes their interaction, including the birth and annihilation of elementary particles. Under “quantum field theory” in this case means a very specific term. This is an area of physics that studies the behavior of quantum fields.

The Dirac Equation in cylindrical coordinates

First let, what is cylindrical coordinate system. Instead of the usual three mutually perpendicular axes to determine the exact location of the point in space use the angle, radius and height. This is the same as the polar coordinate system on the plane, only the addition of a third dimension-height. This system is useful when you want to describe or explore some surface symmetrical about one axis. For quantum mechanics this is a useful and handy tool which allows to significantly reduce the size of formulas and the number of computations. This is a consequence of the axi-symmetric option of the electron cloud in the atom. The Dirac equation in cylindrical coordinates is solved differently than in the conventional system, and sometimes yields unexpected results. For example, some application tasks to determine the behavior of elementary particles (most often electrons) in the quantized field was solved by converting the equation to the cylindrical coordinates.

Using the equation for determining the structure of particle

This equation describes simple particles: those which do not consist of even smaller elements. Modern science is able to measure magnetic moments with high precision. Thus, the mismatch calculated with the Dirac equation the values of the experimentally measured magnetic moment will be to indirectly testify about the complex structure of the particles. Recall that this equation applies to fermions, their spin is half-integer. With the help of this equation was confirmed by the complex structure of protons and neutrons. Each of them consists of even smaller units called quarks. Gluon field holding 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 did not have enough technical power to check this out.