This article describes the wave function and its physical meaning. The application of this concept in the framework of the Schrödinger equation is also considered.
Science is on the verge of discovering quantum physics
At the end of the nineteenth century, young people who wanted to connect their lives with science were discouraged from becoming physicists. There was an opinion that all phenomena have already been discovered and there can no longer be great breakthroughs in this area. Now, despite the seeming completeness of human knowledge, no one will dare to speak in this way. Because this happens often: a phenomenon or effect is predicted theoretically, but people do not have enough technical and technological power to prove or disprove them. For example, Einstein predicted gravitational waves more than a hundred years ago, but it only became possible to prove their existence a year ago. This also applies to the world of subatomic particles (namely, such a concept as a wave function applies to them): until scientists realized that the structure of the atom is complex, they did not need to study the behavior of such small objects.
Spectra and photography
Push todevelopment of quantum physics was the development of photography techniques. Until the beginning of the twentieth century, capturing images was cumbersome, time-consuming and expensive: the camera weighed tens of kilograms, and the models had to stand for half an hour in one position. In addition, the slightest mistake in handling fragile glass plates coated with a photosensitive emulsion led to an irreversible loss of information. But gradually the devices became lighter, the shutter speed - less and less, and the receipt of prints - more and more perfect. And finally, it became possible to obtain a spectrum of different substances. The questions and inconsistencies that arose in the first theories about the nature of spectra 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.
Particle-wave duality
After determining the structure of the atom, the question arose: why does the electron not fall on the nucleus? After all, according to Maxwell's equations, any moving charged particle radiates, therefore, loses energy. If this were the case for the electrons in the nucleus, the universe as we know it would not last long. Recall that our goal is the wave function and its statistical meaning.
An ingenious conjecture of scientists came to the rescue: elementary particles are both waves and particles (corpuscles). Their properties are both mass with momentum and wavelength with frequency. In addition, due to the presence of two previously incompatible properties, elementary particles have acquired new characteristics.
One of them is a hard to imagine spin. In the worldsmaller particles, quarks, there are so many of these properties that they are given absolutely incredible names: flavor, color. If the reader encounters 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.
Schrödinger equation
Find the state in which an elementary particle (and, in a generalized form, a quantum system) is located, allows Erwin Schrödinger's equation:
i ħ[(d/dt) Ψ]=Ĥ ψ.
The designations in this ratio are as follows:
- ħ=h/2 π, where h is Planck's constant.
- Ĥ – Hamiltonian, total energy operator of the system.
- Ψ is the wave function.
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 under consideration.
The concepts of quantum physics
Let the reader not be deceived by the seeming simplicity of the terms used. Words and expressions such as "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 form of the wave function, but this article is of a review nature. For a deeper understanding of this concept, it is necessary to study the mathematical apparatus at a certain level.
Wave function
Her mathematical expressionhas the form
|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, so sometimes it is also called the psi-function.
First you need to understand that the function depends on all coordinates and time. So Ψ(x, t) is actually Ψ(x1, x2… x, t). An important note, since the solution of the Schrödinger equation depends on the coordinates.
Next, it is necessary to clarify that |x> means the basis vector of the selected coordinate system. That is, depending on what exactly needs to be obtained, the momentum or probability |x> will look like | x1, x2, …, x >. Obviously, n will also depend on the minimum vector basis of the chosen system. That is, in the usual three-dimensional space n=3. For the inexperienced reader, let us explain that all these icons near the x indicator are not just a whim, but a specific mathematical operation. It will not be possible to understand it without the most complex mathematical calculations, so we sincerely hope that those who are interested will find out its meaning for themselves.
Finally, it is necessary to explain that Ψ(x, t)=.
Physical essence of the wave function
Despite the basic value of this quantity, it itself does not have a phenomenon or concept as its basis. The physical meaning of the wave function is the square of its total modulus. The formula looks like this:
|Ψ (x1, x2, …, x , t)| 2=ω, where ω is the value of the probability density. In the case of discrete spectra (rather than continuous ones), this value becomes simply a probability.
Consequence of the physical meaning of the wave function
Such a physical meaning has far-reaching implications 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 in orbits around the atomic nucleus.
Let's take two types of hybridization of electrons in atoms with the simplest forms of clouds: s and p. Clouds of the first type are spherical in shape. But if the reader remembers from textbooks on physics, these electron clouds are always depicted as some kind of blurry cluster of points, and not as a smooth sphere. This means that at a certain distance from the nucleus there is a zone with the highest probability of encountering an 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 complex surface on which the probability of finding an electron is the highest. But even close to this “dumbbell”, both farther and closer to the core, such a probability is not equal to zero.
Normalization of the wave function
The latter implies the need to normalize the wave function. By normalization is meant such a "fitting" of some parameters, in which it is truesome ratio. If we consider spatial coordinates, then the probability of finding a given particle (an electron, for example) in the existing Universe should be equal to 1. The formula looks like this:
ʃV Ψ Ψ dV=1.
Thus, the law of conservation of energy is fulfilled: 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. And it doesn't matter if this particle is inside a star or in a giant cosmic void, it has to be somewhere.
A little higher we mentioned that the variables on which the function depends can also be non-spatial coordinates. In this case, normalization is carried out over all parameters on which the function depends.
Instant travel: trick or reality?
In quantum mechanics, separating mathematics from physical meaning is incredibly difficult. For example, the 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 the quantum state of the system changes instantly during the measurement. This phenomenon is usually referred to as the reduction or collapse of the wave function. If this is possible in reality, quantum systems are capable of moving at infinite speed. But the speed limit for real objects of our Universeimmutable: nothing can travel faster than light. This phenomenon has never been recorded, but it has not yet been possible to refute it theoretically. Over time, perhaps, this paradox will be resolved: either humanity will have a tool that will fix such a phenomenon, or there will be a mathematical trick that will prove the inconsistency 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.
Wave function of a multiparticle system (hydrogen atom)
As we have stated throughout the article, the psi-function describes one elementary particle. But on closer inspection, the hydrogen atom looks like a system of just two particles (one negative electron and one positive proton). The wave functions of the hydrogen atom can be described as two-particle or by a density matrix type operator. These matrices are not exactly an extension of the psi function. Rather, they show the correspondence between the probabilities of finding a particle in one and the other state. It is important to remember that the problem is solved only for two bodies at the same 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, an incredible similarity can be traced between the most "rough" mechanics and very "fine" quantum physics. Therefore, one should not think that since quantum mechanics exists, new ideas cannot arise in ordinary physics. The interesting is hidden behind anyby turning mathematical manipulations.
