This article describes the wave function and its physical meaning. The use of this concept in the framework of the Schrödinger equation is also considered.
In the late nineteenth century young peoplewho wanted to connect their lives with science, discouraged becoming physicists. There was an opinion that all phenomena are already open and there can no longer be great breakthroughs in this area. Now, despite the apparent fullness of the knowledge of mankind, no one will dare say this way. Because it happens so often: the phenomenon or effect is predicted theoretically, but people lack the technical and technological power to prove or disprove them. For example, Einstein predicted gravitational waves more than a hundred years ago, but it became possible to prove their existence only a year ago. This also applies to the world of subatomic particles (namely, such a concept as the wave function applies): until scientists understood that the structure of the atom is complex, they did not have to study the behavior of such small objects.
The impetus for the development of quantum physics wasdevelopment of photography techniques. Until the beginning of the twentieth century, capturing images was cumbersome, lengthy and expensive: the camera weighed tens of kilograms, and the models had to stand for half an hour in the same pose. In addition, the slightest mistake in handling fragile glass plates coated with a photosensitive emulsion led to irreversible loss of information. But gradually the devices became lighter, the shutter speed - less and less, and getting prints - more and more perfect. Finally, it became possible to obtain a spectrum of different substances. Questions and inconsistencies that arose in the first theories about the nature of the spectra, and gave rise to a whole new science. The wave function of a particle and its Schrödinger equation became the basis for the mathematical description of the behavior of the microworld.
After determining the structure of the atom, the question arose:why the electron does not fall on the nucleus? After all, according to Maxwell's equations, any moving charged particle emits, therefore, loses energy. If this were so for electrons in the nucleus, the known universe would not exist for long. Recall that our goal is the wave function and its statistical meaning.
A brilliant conjecture of scientists came to the rescue:elementary particles are both waves and particles (corpuscles) at the same time. Their properties are mass with impulse, and wavelength with frequency. In addition, due to the presence of two previously incompatible properties, elementary particles acquired new characteristics.
One of them is hard to imagine spin.In the world of smaller particles, quarks, these properties are so numerous that they are given absolutely unbelievable names: aroma, color. If the reader meets them in a book on quantum mechanics, let him remember: they are not at all what they seem at first glance. However, how to describe the behavior of such a system, where all elements have a strange set of properties? The answer is in the next section.
Find the state in which the elementary particle is located (and in a generalized form, and the quantum system), allows the equation of Erwin Schrödinger:
i ħ [(d / dt) Ψ] = ψ.
The designations in this relationship are as follows:
By changing the coordinates in which this function is solved and the conditions in accordance with the type of particle and the field in which it is located, one can obtain the law of behavior of the system in question.
Пусть читатель не обольщается кажущейся простотой terms used. Words and expressions like “operator”, “total energy”, “unit cell” are physical terms. Their values should be clarified separately, and it is better to use textbooks. Next, we will give a description and a view of the wave function, but this article is of an overview nature. For a deeper understanding of this concept, it is necessary to study the mathematical apparatus at a certain level.
Its mathematical expression is
| ψ (t)> = ʃ Ψ (x, t) | x> dx.
The wave function of an electron or any other elementary particle is always described by the Greek letter Ψ, therefore, sometimes it is also called the psi function.
First you need to understand that the function depends on all coordinates and time. That is, Ψ (x, t) is actually Ψ (x1, x2... xMr., t). An important remark, since the solution of the Schrödinger equation depends on the coordinates.
Further, it is necessary to clarify that under | x>implies the base vector of the selected coordinate system. That is, depending on what needs to be received, the impulse or probability | x> will have the form | x1, x2,…, XMr.>.Obviously, n will also depend on the minimal vector basis of the chosen system. That is, in the usual three-dimensional space, n = 3. For an inexperienced reader, let us explain that all these icons near the indicator x are not just a whim, but a specific mathematical operation. It will not be possible to understand it without the most complicated mathematical calculations, so we sincerely hope that those interested will find out its meaning for themselves.
Finally, it is necessary to explain that Ψ (x, t) =
Despite the basic value of this quantity, it itself does not have a phenomenon or a concept at the base. The physical meaning of the wave function is the square of its full modulus. The formula looks like this:
| Ψ (x1, x2,…, XMr., t) |2= ω,
where ω is the probability density value. In the case of discrete spectra (rather than continuous), this quantity acquires the value of mere probability.
This physical meaning has far reachingimplications for the entire quantum world. As it becomes clear from the value of ω, all states of elementary particles acquire a probabilistic hue. The most obvious example is the spatial distribution of electron clouds at orbitals around the atomic nucleus.
Take two types of electron hybridization in atomswith the most simple cloud shapes: s and p. Clouds of the first type have the form of a ball. But if the reader remembers from the textbooks on physics, these electron clouds are always depicted as a kind of vague cluster of points, and not as a smooth sphere. This means that at a certain distance from the nucleus, the zone is most likely to meet the s-electron. However, a little closer and a little further this probability is not zero, it is just less. In this case, for p-electrons, the shape of the electron cloud is depicted as a somewhat blurry dumbbell. That is, there is a rather complicated surface on which the probability of finding an electron is the highest. But even close to this “dumbbell” both further and closer to the core, this probability is not zero.
The latter should be normalized.wave function. Under the normalization refers to such a "fit" of some parameters, in which some relationship is true. If we consider the spatial coordinates, then the probability of finding a given particle (electron, for example) in the existing Universe should be equal to 1. The formula looks like this:
ʃAT Ψ * Ψ dV = 1.
Таким образом, выполняется закон сохранения energy: if we are looking for a specific electron, it must be entirely in a given space. Otherwise, solving the Schrödinger equation simply does not make sense. It does not matter if this particle is inside a star or in a giant cosmic entry, it must be somewhere.
Slightly above, we mentioned that the variables on which the function depends may be non-spatial coordinates. In this case, the normalization is carried out for all parameters on which the function depends.
In quantum mechanics, separate math fromPhysical sense is incredibly difficult. For example, a quantum was introduced by Planck for the convenience of the mathematical expression of one of the equations. Now the principle of discreteness of many quantities and concepts (energy, angular momentum, field) underlies the modern approach to the study of the microworld. Ψ also has this paradox. According to one of the solutions of the Schrödinger equation, it is possible that, when measured, the quantum state of the system changes instantaneously. This phenomenon is usually referred to as the reduction or collapse of the wave function. If this is possible in reality, quantum systems can move at infinite speed. But the speed limit for real objects of our Universe is immutable: nothing can move faster than light. This phenomenon has never been fixed, but theoretically it has not been possible to refute it. Over time, perhaps this paradox will be resolved: either humanity will have a tool that will fix such a phenomenon, or a mathematical trick will be found that will prove the failure of this assumption. There is a third option: people will create such a phenomenon, but at the same time the Solar system will fall into an artificial black hole.
As we argued throughout the paper,psi function describes one elementary particle. But upon closer inspection, a hydrogen atom resembles a system of only two particles (one negative electron and one positive proton). The wave functions of a hydrogen atom can be described as two-particle or by an operator such as a density matrix. These matrices are not exactly the continuation of the psi function. They rather show the correspondence of probabilities of finding a particle in one and another state. It is important to remember that the problem is solved only for two bodies at a time. Density matrices are applicable to pairs of particles, but are not possible for more complex systems, for example, when three or more bodies interact. In this fact, there is an incredible similarity between the most "rough" mechanics and very "subtle" quantum physics. Therefore, one should not think that since quantum mechanics exists, in ordinary physics there can be no new ideas. Interesting is hidden behind any turn of mathematical manipulations.
