The uncertainty relations is the mathematically formulated principle of the quantum theory, according to which the existence of such states of a physical system in which two dynamical variables (hereafter denoted in general A and B) are forbidden would have a definite value if these variables are canonically conjugate quantities. Since several different pairs of canonically conjugate quantities can take place, it is possible to speak in the plural about the uncertainty relations. Although the uncertainty relations is considered as the principle of quantum mechanics, its action can be traced from the concepts of classical mechanics. The canonically conjugate quantities are mathematical variables included in the so-called ” canonical equations of mechanics (Hamilton’s equations) and determining the state of a mechanical system at any time. As usually canonically conjugate variables, generalized coordinates Q and generalized momenta P are chosen. With the help of the so-called canonical transformations, we can pass from Q and P to other canonically conjugate quantities Q and P, which may have a different physical meaning.

If two variables A and B are canonically conjugate to each other in the sense of the Hamiltonian formalism, then no experiment can lead to the simultaneous accurate measurement of such variables. The inaccuracy of the measurement is connected not with the imperfection of the measuring technique, but with the objective properties of the system under investigation. Mathematically, the uncertainty relations are written in the general form as follows: ΔΑ · ΔΒ≥h. This entry means that the product of the measurement errors of canonically conjugate quantities can not be of the order of magnitude less than the Planck constant H. The more accurately the value of one of the values entering into the ratio is determined, the less definitely the value of the other quantity: when trying to determine the value of one of the values in an extremely precise way, the uncertainty of the value of the other turns out to be in the region of infinite values. Taking into account the extremely small Planck constant H in comparison with macroscopic quantities of the same physical dimension, it is necessary to conclude that the uncertainty relations are significant only in the study of phenomena of atomic scale.

The mathematical expression of the uncertainty relations was first formulated by V. Heisenberg in 1927 in the context of the problem of the paradoxical combination of wave and particle properties in microparticles. Discussing this problem with him, H. Bohr persistently sought a way to combine corpuscular and wave properties in microworld objects rationally. Reflecting on theoretical and cognitive problems, Bohr then came to the idea of complementarity – corpuscular and wave properties do not exclude each other but are in a complementary relationship. Sometimes the Bohr complementarity principle is represented as a generalization of the uncertainty relations.

However, initially, Heisenberg strongly denied the possibility of such a construction of a new theory, in which the wave properties of particles would be taken into account. He was then convinced that it was possible to build a new theory solely on the basis of the idea of discreteness. He formulated the specific uncertainty relations between the coordinate and momentum of a particle under the influence of Bohr, who considered it necessary to find an expression for characterizing the relationships between particle and wave properties of microparticles. In the uncertainty relation, various additional properties of particles are uniquely combined in one formula – on the basis of the methodological principle; a special mathematical apparatus was constructed.

A distinctive feature of atomic processes is their corpuscular-wave nature, which is manifested in experiments. The particle motion is associated with the propagation of a specific wave, and the particle itself can be detected at any point of this wave. As a result, the motion of the microparticle has a probabilistic character. For example, in an experiment where the phenomenon of electron diffraction is being studied, a particle of a certain energy falls on a diffraction grating; the process of falling off an electron is repeated many times. This gives rise to a characteristic diffraction pattern that indicates the wave properties of an electron, for the phenomenon of diffraction, consists precisely in the deviation from the rectilinear motion inherent in the laws of geometric optics, which is abstracted from the wave character of the physical process.

The picture of the diffraction of an electron shows that in the act of interaction of an electron with a diffraction grating all its cells participate. This means that it is impossible to predict the trajectory of the motion of an electron when it falls on a lattice, in other words, it is impossible to know in which direction the electron will move. The observed phenomenon of electron diffraction confirms the wave nature of microparticles and, at the same time, indicates the probabilistic nature of their behavior. In quantum theory, the state of a particle in the described situation is expressed by a wave function and can not be represented with an accuracy characteristic of classical concepts. Thus, classical concepts of momentum and coordinates are not applicable to microscopic objects. In describing the behavior of microparticles, it becomes necessary to take into account their quantum properties, which is manifested in the uncertainty relations.