Все науки. №7, 2024. Международный научный журнал

UDK: 511.24

Ibratjon Aliyev1, Sultonali Abdurakhmonov2, Erkinjon Kholmatov2, Nurmakhamad Juraev3, Mamatisa Djalilov3

1SRI «PRNR», Electron Laboratory LLC, 151100, Republic of Uzbekistan, Ferghana region, Margilan

2Fergana Polytechnic Institute, 150100, Republic of Uzbekistan, Ferghana region, Ferghana

3Fergana branch of Tashkent University of Information Technologies named after Mukhammad al-Khwarizmi, 185, Mustaqillik street, Fergana, 150118, Uzbekistan

Abstract. The paper presents a study on modeling the quantum mechanical process of tunneling a beam of charged particles to transmit information over long distances. The Schrodinger equation is used for the analysis, boundary and initial conditions are formulated. The initial conditions are the values of the quantum mechanical probability function from the square of its modulus at the initial moment of time and at the final moment, depending on the distance. Experimental data were used as data for the calculation. The solution of the problems was carried out using the method of separation of Fourier variables. In conclusion, the parameters of the simulated system with its features and corresponding graphical representations are given. Based on the results obtained, conclusions are drawn on the effect of tunneling in the transmission of information.

Key words: Subsequent, spirit, Kuiper, equation, planet Earth, transmission technology

Introduction

The development of information technology in the modern industry leads to the need to improve data transmission systems at high speeds. The improvement of technologies for sending electromagnetic signals between different communication systems was initially organized on the principles of interaction through a direct conductor, which was observed in local installations where individual blocks of a particular design interacted with each other transmitting the necessary data [1]. In this case, the speed and volume of data transmission was limited by the network capabilities, in the case of speed, it was the speed depending on the difference in the created potentials or on the speed of charges in the conductor, the volume depended on the quantitative possibility of transferring charges over a certain distance.

Subsequent developments led to the discovery of oscillatory circuit technology, and even taking into account the development of the original technology with the search for combining materials, the method of transmitting information by direct transmission through an electromagnetic field at a speed scale became and remained a priority [2—4]. The boundary value in this case was the velocity of wave propagation, depending on the parameters of the medium, equal to the speed of light in a particular medium. Bandwidth also became the final indicator of volume, but unlike the first option, a significant obstacle appeared in this data transmission technology in the global and local sense – data loss. In this case, the opening of opportunities for third-party perception of information or decryption is not understood, but the direct loss of data due to a decrease in the amplitude and power of the directed electromagnetic signal towards the receiver.

With the development of the basics of technology and the improvement of both methods of information transmission, a third class of technologies was formed [2—3; 5—6] capable of sending signals over long distances with minimal losses and at a speed equal in magnitude to the speed of propagation of an electromagnetic field in a vacuum – fiber optic networks. At the same time, they had all the advantages of conductor technology regarding the possibility of increasing volumes at maximum speed, however, in this case, the speed, even taking into account its extreme indicators with the achievement of the maximum vacuum level, was insufficient with the growing needs of current technological networks. An increase in the volume of data transmission leads to an increase in the scale of installations, which does not meet the requirements of the current time, based on this, there is a need for research and development of technologies with more advanced capabilities.

With the development of quantum theory, the attention of researchers began to be attracted by the effect of quantum tunneling, now used in tunnel diodes, computer networks, tunnel microscope [4; 7], superconducting systems and many other technologies. The study theoretically did not consider new effects associated with the accelerated transfer of information between objects, therefore, the theoretical justification of the tunneling effect in the transmission of information is relevant.

Materials and methods of research

The research materials of the external probes, their parameters, and information about the studies carried out were used for the study. The methods used in the research were the method of analysis, classification, and theoretical modeling using partial differential equations.

Equations and mathematics

Initially, it is important to determine the class of tasks where the use of tunneling technology may be in demand. Due to the fact that the maximum transmission speed when using fiber—optic systems or oscillatory circuits with a transmitting electromagnetic field is the speed of light in the specified medium and in the maximum case, it is 299,792,458 m/s, capable of circumnavigating the planet Earth in 0.134 s – the maximum possible delay in the transmission system, it becomes obvious that the scope of application data transmission systems with high speeds are becoming cosmic in scale. Space probes are already being used by international organizations. The earliest probes are Pioneer 5, 6 (A), 7 (B), 8 (C) and others, the earliest are Helios A, Helios B, ISEE—3, Ulysses, Wind, SOHO, ACE and modern ones are the second exit of Ulysses, Genesis [3—7; 11—12; 14].

Also, among the probes there are Stereo A operating to date since 2006, DSCOVR since 2015, Parker Solar Probe until December 2025, ESA from 2020, ISRO from 2023 and others. Also, each of the probes is divided into different categories – Solar probes aimed at exploring the Sun and located at a comparative distance from Earth up to 1 astronomical unit, which also include Mercury probes – Mariner 10, MESSENGER, BeliKolombo and others and Venus probes – Venus 1, Mariner 1, Sputnik 19, Sputnik 2, Cosmos 27, Zone 1, Venus 8, Mariner 10, Pioneer Venus, Venus 12, Venus 11, Magellan, Galileo, Cassini and others [2—5; 7—14]. But there are also probes directed in the opposite direction, which include the Mars probes – Mars 1B No. 1, Mariner 3, Zone 2, Mars 1969A, Cosmos 419, PrOP-M, Mars 5, Mars 6, Mars 7, Phobos 1 and 2, the 2001 Martian Odyssey, Nozomi, Mars Express not counting a large number of rovers – Mars A «Spirit», Mars B «Opportunity», along with others – Rosetta, Phoenix, Dawn, Marco A «WALL—I», Marco B «Eve», Tianwen-1, Zhuron, Psyche, Hera, Europa Clipper and others [13—14; 15].

These include the probes of the Mars satellite, as well as other satellites and even asteroids – Dawn, Galileo, who visited asteroids 951 Gaspra, 243 Ida, Clementine, who visited 1620 Geographos, among others. Separately, there are probes of the largest planet – Jupiter in the person of Pioneer 10, Pioneer 11, Cassini, Ulysses, New Horizons, Galileo, Juno, SOC, and its satellites, in particular Ganymede. Also, Saturn – Voyager 1, Voyager 2, Cassini, and its moon Titan – Huygens [7—8; 14—15]. And even the most distant planets of the Solar System – Uranus, Neptune, which Voyager 1 was able to reach, and even the dwarf planet Pluto, which the New Horizons probe was able to reach, which continued its journey and was able to reach the Kuiper belt to the space object 486958 Arrokot, located at a distance of 43.4 astronomical units from the Sun [1—8; 9]!

Based on the particular list of space objects of artificial and research origin available in space, it becomes obvious that the most important scientific research data on a wide variety of objects in and outside the Solar System must be transmitted at the highest possible speeds. At the moment, taking into account the available technologies, at a distance of 43.4 astronomical units, taking into account the distance to Earth of 42.4 astronomical units from the farthest probe New Horizons, the signal reaches in 21,157.8 seconds or 5 hours 52 minutes and 37.8 seconds.

In order to be able to transmit a signal over a distance using tunneling technology, it is necessary to initially present the information in the form of a group of charges, a beam, to which additional energy is transmitted. It can be transmitted by means of an electric field in a small portable accelerator, where, upon reaching the required defect, the particle will tunnel at a speed several times higher than the speed of light, according to the nature of the phenomenon of quantum tunneling to Earth at a specified point with a certain error and degree of loss.

But for a better consideration of the issue, it is necessary to proceed to the formulation of the appropriate task.

Setting the task

Initially, it is known that the phenomenon is described by the Schrodinger equation in the dynamic representation (1)

The equation involves a potential barrier U (x, t), which is a variable function and depends on multiple parameters – objects that are located between the source and the receiver, which must be overcome by the guided particle. Most often, vacuum prevails in outer space, but there are also quite a few obstacles, for the initial idea, we define the value of the potential barrier as the amount of energy of the mass of all matter located on the communication channel, with an average density of matter in the Solar system. Thus, the communication channel is an imaginary cylinder with a radius equal to the radius of a beam of 10 microns, 42.4 astronomical units long, from which the corresponding volume and mass are calculated, taking into account the average density of matter in the solar system of 0.931*10—26 kg/m3 (2).

And also, based on the height of the potential barrier, it is possible to determine the boundary and initial conditions for the state function of the quantum mechanical system in (3).

The boundary conditions are formed from several statements. For use, such a system is necessary in which the probability of finding the beam after sending it in the radiator should be zero, at the specified target – on the Ground, should be equal to 100 percent. Despite the fact that with classical propagation at a time, taking into account such distance measurement, it is necessary that after 21,157,80283 seconds-meters, the probability of finding the beam on Earth was 100% and zero when reaching the Sun at 21,656,80762 seconds-meters. Based on the obtained indicators with respect to one dimension and time, equation (1) can be solved with the specified boundary conditions (3).

To do this, the Fourier variable separation method will be used, with respect to solving an equation of the form (4), the form of the function (5) will be adopted, where, after substitution, the form (6) is formed, from which 2 separate ordinary differential equations are derived.

The equation is solved in time to the state of the general form, according to (7), but due to the presence of initial conditions in (3), the present form can be solved by means of representation in the form of a system (8), taking into account the finding of the formula-dependence on the independent variables of the general form of the function (9), where after solving the formed equation after substituting the formula of the independent variable, the form of the introduced constant (10) in (6) is formed.

The value of the constant makes it possible to determine the value of the first and, accordingly, the opposite of the second independent constant (11), which, after substituting into the general form of the time function (7), gives its private emerging form (12) and (Fig. 1).

Fig. 1. Graph of the function

To continue the study, after establishing the actual form of the function in time, it is necessary to solve the formed ordinary differential equation with respect to the coordinate, which was obtained in the ratio (6). Since the value for the constant was also obtained in (10), after substitution, a final form of an ordinary differential equation in coordinate is formed, for which there is a constant from the characteristic of the form (13), and then the general form of the function (14).

When forming the problem, boundary conditions were also indicated, the substitution of which allows us to operate from the conclusions of the expression for the third and also opposite, as can be seen from the boundary conditions, independent constant, which in this case has a large-scale appearance, for which a replacement is introduced (15).

Substituting the resulting replacement allows you to ultimately form a simplified view of the function at the coordinate (16).

The result of the study is the assembled form of the function of the quantum mechanical state of the tunneling system. However, in this case, the desired function is initially a probability distribution function, for which only the square of its module has physical meaning. Based on this, it is possible to form the form of the square of the module of the function, as for the boundary conditions to be set, for the reason that initially the problem required complete preservation of information and in the case of data transition to an imaginary space, the boundary conditions were set to the original form of the function, for which the square of the module is represented in (17—18).

Thus, a full-fledged pattern was derived that predicts the probability distribution of finding a response signal for a given case at the maximum distance from the Earth for the probe.

Results

According to the results of the study, a function depending on 2 variables was obtained, which can be constructed and according to the initial projection, the first graph of the function is formed without an additional power factor. After applying the appropriate processing (18), the function remains the same, but its parameters describe the actual picture (Fig. 2).

Fig. 2. Graph of the probability distribution of finding a signal beam

So, from Figure 2 it is clearly seen that as time passes along the numerical axis in increments of 2.35022802 seconds-meters to 70.51 seconds-meters or along the ordinary axis in increments of the same amount of meters to 30.55296 meters of range, it is possible to state the probability of finding a beam at 11.27337%, which makes it possible to assert, based on The parameter of 70.51 seconds is meters or 21,137,419,062 meters in the usual metric coordinate system, which means that the beam is likely to be found near the Earth and the signal can be received accordingly.

Speaking of losses, it is enough to use the expression for the barrier passage coefficient (18) and its graph (Fig. 3) as the initial beam energy approaches the value of the potential barrier, determined relative to the case under study in (2), where it is also important to note that in all cases the information carrier or The information about matter was electrons, for protons, deuterons, tritons, other ions, quarks, mesons and other particles, such decompositions will have a different character.

Fig. 3. Percentage of information loss during tunneling

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