Wave function and its statistical meaning. Types of wave function and its collapse

This article describes the wave function and its physical meaning. We also consider the application of this concept in the framework of the Schrödinger equation.

Science on the Threshold of the Discovery of Quantum Physics

At the end of the nineteenth century young people who wanted to link their lives with science were discouraged from becoming physicists. There was an opinion that all phenomena are already open and there can not be great breakthroughs in this field. Now, in spite of the apparent fullness of the knowledge of mankind, no one will dare say this way. Because it happens often: a 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 was possible to prove their existence only a year ago. This also applies to the world of subatomic particles (namely, the term wave function is applicable to them): until scientists understood that the structure of the atom is complex, they did not have to study the behavior of such small objects.

Spectra and photography

The impetus to the development of quantum physics was the development of photography techniques. Until the beginning of the twentieth century, capturing images was cumbersome, long and expensive: the camera weighed tens of kilograms, and the models had to stand for half an hour in one pose. In addition, the slightest error in handling fragile glass plates coated with a photosensitive emulsion resulted in irreversible loss of information. But gradually the apparatus became more and more easy, the exposure - all less, and the receipt of prints - all 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 generated a whole new science. The wave function of the particle and its Schrödinger equation became the basis for the mathematical description of the behavior of the microworld.

Corpuscular Wave Dualism

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 emits, therefore, loses energy. If this were so for the electrons in the nucleus, the universe known to us would not last long. Recall, our goal is the wave function and its statistical meaning.

The brilliant guess of scientists came to the rescue: elementary particles are simultaneously 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 difficultly represented spin. In the world of smaller particles, quarks, these properties are so many that they are given absolutely incredible 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 can we describe the behavior of such a system, where all elements have a strange set of properties? The answer is in the next section.

The Schrodinger equation

To find the state in which the elementary particle is located (and in the generalized form also the quantum system), the Erwin Schrödinger equation can be used :

I ħ [(d / dt) Ψ] = Ĥ ψ.

The notation in this relation is as follows:

• Ħ = h / 2 π, where h is the Planck constant.
• Ĥ - Hamiltonian, the operator of the total energy of the system.
• Ψ is the wave function.

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 under consideration.

The concepts of quantum physics

Let the reader not be deluded by the apparent simplicity of the terms used. Such words and expressions as "operator", "total energy", "unit cell" are physical terms. Their values should be specified separately, and it is better to use textbooks. Next, we give a description and the form 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.

Wave function

Its mathematical expression has 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 the coordinates and time. That is, Ψ (x, t) is actually Ψ (x ₁ , x ₂ ... x _n , t). An important remark, since the solution of the Schrodinger equation depends on the coordinates.

Next, it is necessary to explain that by | x> we mean the basis vector of the chosen coordinate system. That is, depending on what it is necessary to obtain, the momentum or probability | x> will have the form | X ₁ , x ₂ , ..., x _n >. 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 exponent x are not just a whim, but a concrete mathematical action. To understand it without complicated mathematical calculations will not succeed, therefore we sincerely hope that those interested will find out its meaning.

Finally, it is necessary to explain that Ψ (x, t) = .

The physical essence of the wave function

Despite the basic significance of this magnitude, it does not have a phenomenon or concept at its base. The physical meaning of the wave function lies in the square of its full modulus. The formula looks like this:

| Ψ (x ₁ , x ₂ , ..., x _n , t) | ² = ω,

Where ω has a probability density. In the case of discrete spectra (and not continuous ones), this quantity acquires a value of simply probability.

Corollary of the physical meaning of the wave function

Such a physical meaning has far-reaching consequences for the entire quantum world. As it becomes clear from the value of ω, all states of elementary particles acquire a probabilistic shade. The most obvious example is the spatial distribution of electron clouds on orbitals around the atomic nucleus.

Let us take two types of hybridization of electrons in atoms with the simplest forms of clouds: s and p. Clouds of the first type have the form of a sphere. But if the reader remembers from the textbooks on physics, these electronic clouds are always portrayed as some vague 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 greatest probability of encountering an s-electron. However, a little closer and a little further this probability is not zero, it's just smaller. In this case, for p-electrons, the shape of the electron cloud is represented in the form of a somewhat diffuse dumbbell. That is, there is a rather complex surface on which the probability to find the electron is the highest. But even near this "dumbbell" both further and closer to the nucleus such probability is not equal to zero.

Normalization of the wave function

It follows from the latter that it is necessary to normalize the wave function. By normalization we mean such a "fitting" of certain parameters, for which a certain relation holds. 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:

Ʃ _V 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 the given space. Otherwise, solving the Schrödinger equation simply does not make sense. And it does not matter whether this particle is inside a star or in a giant cosmic hollow, it must be somewhere.

A little bit earlier we mentioned that the variables on which the function depends can also be non-spatial coordinates. In this case, the normalization is performed over all parameters from which the function depends.

Instant movement: reception or reality?

In quantum mechanics, it is incredibly difficult to separate mathematics from physical meaning. 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. We also have such a paradox. According to one of the solutions of the Schrödinger equation, it is possible that in measuring 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 are able to move with infinite speed. But the limitation of velocities for the real objects of our Universe is immutable: nothing can move faster than light. This phenomenon has not been recorded once, but it has not been theoretically possible to refute it. In time, perhaps, this paradox will be resolved: either humanity will have a tool that fixes this phenomenon, or there is a mathematical trick that proves the inconsistency of this assumption. There is a third option: people will create such a phenomenon, but the Solar system will fall into an artificial black hole.

The wave function of the multiparticle system (the hydrogen atom)

As we have argued throughout the article, the psi function describes one elementary particle. But upon closer examination, the hydrogen atom is similar to 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 an operator of the density matrix type. These matrices are not exactly the continuation of the psi function. They rather show the matching of probabilities to find 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 particle pairs, but are not possible for more complex systems, for example, when three or more bodies interact. In this fact, an incredible similarity exists between the "grossest" mechanics and the very "subtle" quantum physics. Therefore, we should not think that since quantum mechanics exists, in ordinary physics, new ideas can not arise. Interesting lies behind any turn of mathematical manipulations.