There is a possibility that a quantum particle will penetrate a barrier that is insurmountable for a classical elementary particle.
Imagine a ball rolling inside a spherical hole dug in the ground. At any moment of time, the energy of the ball is distributed between its kinetic energy and the potential energy of gravity in a proportion depending on how high the ball is relative to the bottom of the hole (according to the first law of thermodynamics). When the ball reaches the side of the hole, two scenarios are possible. If its total energy exceeds the potential energy of the gravitational field, determined by the height of the ball's location, it will jump out of the hole. If the total energy of the ball is less than the potential energy of gravity at the level of the side of the hole, the ball will roll down, back into the hole, towards the opposite side; at the moment when the potential energy is equal to the total energy of the ball, it will stop and roll back. In the second case, the ball will never roll out of the hole unless additional kinetic energy is given to it - for example, by pushing it. According to Newton's laws of mechanics, the ball will never leave the hole without giving it additional momentum if it does not have enough of its own energy to roll overboard.
Now imagine that the sides of the pit rise above the surface of the earth (like lunar craters). If the ball manages to fall over the raised side of such a hole, it will roll further. It is important to remember that in the Newtonian world of the ball and the hole, the fact that the ball will roll further over the side of the hole has no meaning if the ball does not have enough kinetic energy to reach the top edge. If it does not reach the edge, it simply will not get out of the hole and, accordingly, under no conditions, at any speed and will not roll anywhere further, no matter how high above the surface the edge of the side is located outside.
In the world of quantum mechanics, things are different. Let's imagine that there is a quantum particle in something like such a hole. In this case, we are no longer talking about a real physical hole, but about a conditional situation when a particle requires a certain supply of energy necessary to overcome the barrier that prevents it from breaking out of what physicists have agreed to call "potential hole". This pit also has an energy analogue of the side - the so-called "potential barrier". So, if outside the potential barrier the level of energy field intensity is lower than the energy possessed by the particle, it has a chance to be “overboard”, even if the real kinetic energy of this particle is not enough to “go over” the edge of the board in the Newtonian sense . This mechanism of a particle passing through a potential barrier is called the quantum tunneling effect.
It works like this: in quantum mechanics, a particle is described through a wave function, which is related to the probability of the particle being located in a given place at a given moment in time. If a particle collides with a potential barrier, Schrödinger's equation allows us to calculate the probability of the particle penetrating through it, since the wave function is not just energetically absorbed by the barrier, but is extinguished very quickly - exponentially. In other words, the potential barrier in the world of quantum mechanics is blurred. It, of course, prevents the particle from moving, but is not a solid, impenetrable boundary, as is the case in classical Newtonian mechanics.
If the barrier is low enough or if the total energy of the particle is close to the threshold, the wave function, although it decreases rapidly as the particle approaches the edge of the barrier, leaves it a chance to overcome it. That is, there is a certain probability that the particle will be detected on the other side of the potential barrier - in the world of Newtonian mechanics this would be impossible. And once the particle has crossed the edge of the barrier (let it have the shape of a lunar crater), it will freely roll down its outer slope away from the hole from which it emerged.
A quantum tunnel junction can be thought of as a kind of "leakage" or "percolation" of a particle through a potential barrier, after which the particle moves away from the barrier. There are plenty of examples of this kind of phenomena in nature, as well as in modern technologies. Take a typical radioactive decay: a heavy nucleus emits an alpha particle consisting of two protons and two neutrons. On the one hand, one can imagine this process in such a way that a heavy nucleus holds an alpha particle inside itself through intranuclear binding forces, just as the ball was held in the hole in our example. However, even if an alpha particle does not have enough free energy to overcome the barrier of intranuclear bonds, there is still a possibility of its separation from the nucleus. And by observing spontaneous alpha emission, we receive experimental confirmation of the reality of the tunnel effect.
Another important example of the tunnel effect is the process of thermonuclear fusion that powers stars (see Evolution of stars). One of the stages of thermonuclear fusion is the collision of two deuterium nuclei (one proton and one neutron each), resulting in the formation of a helium-3 nucleus (two protons and one neutron) and the emission of one neutron. According to Coulomb's law, between two particles with the same charge (in this case, protons that are part of deuterium nuclei) there is a powerful force of mutual repulsion - that is, there is a powerful potential barrier. In Newton's world, deuterium nuclei simply could not come close enough to synthesize a helium nucleus. However, in the depths of stars, the temperature and pressure are so high that the energy of the nuclei approaches the threshold of their fusion (in our sense, the nuclei are almost at the edge of the barrier), as a result of which the tunnel effect begins to operate, thermonuclear fusion occurs - and the stars shine.
Finally, the tunnel effect is already used in practice in electron microscope technology. The action of this tool is based on the fact that the metal tip of the probe approaches the surface under study at an extremely short distance. In this case, the potential barrier prevents electrons from metal atoms from flowing to the surface under study. When moving the probe at an extremely close distance along the surface under study, it seems to be moving atom by atom. When the probe is in close proximity to atoms, the barrier is lower than when the probe passes between them. Accordingly, when the device “gropes” for an atom, the current increases due to increased electron leakage as a result of the tunneling effect, and in the spaces between the atoms the current decreases. This allows for a detailed study of the atomic structures of surfaces, literally “mapping” them. By the way, electron microscopes provide the final confirmation of the atomic theory of the structure of matter.
TUNNEL EFFECT(tunneling) - quantum transition of a system through a region of motion prohibited by classical mechanics. A typical example of such a process is the passage of a particle through potential barrier when her energy
less than the height of the barrier. Particle momentum R in this case, determined from the relation 
Where U(x)- potential particle energy ( T- mass), would be in the region inside the barrier, an imaginary quantity. IN quantum mechanics thanks to uncertainty relationship Between the impulse and the coordinate, subbarrier motion becomes possible. The wave function of a particle in this region decays exponentially, and in the quasiclassical case (see Semiclassical approximation)its amplitude at the point of exit from under the barrier is small.
One of the formulations of problems about the passage of potential. barrier corresponds to the case when a stationary flow of particles falls on the barrier and it is necessary to find the value of the transmitted flow. For such problems, a coefficient is introduced. barrier transparency (tunnel transition coefficient) D, equal to the ratio of the intensities of the transmitted and incident flows. From the time reversibility it follows that the coefficient. Transparencies for transitions in the "forward" and reverse directions are the same. In the one-dimensional case, coefficient. transparency can be written as
integration is carried out over a classically inaccessible region, X 1,2 - turning points determined from the condition At turning points in the classical limit. mechanics, the momentum of the particle becomes zero. Coef. D 0 requires for its definition an exact solution of quantum mechanics. tasks.
If the condition of quasiclassicality is satisfied
along the entire length of the barrier, with the exception of the immediate neighborhoods of turning points x 1.2 coefficient D 0 is slightly different from one. Creatures difference D 0 from unity can be, for example, in cases where the potential curve. energy from one side of the barrier goes so steeply that the quasi-classical the approximation is not applicable there, or when the energy is close to the barrier height (i.e., the exponent expression is small). For a rectangular barrier height U o and width A coefficient transparency is determined by the file
Where 
The base of the barrier corresponds to zero energy. In quasiclassical case D small compared to unity.
Dr. The formulation of the problem of the passage of a particle through a barrier is as follows. Let the particle in the beginning moment in time is in a state close to the so-called. stationary state, which would happen with an impenetrable barrier (for example, with a barrier raised away from potential well to a height greater than the energy of the emitted particle). This state is called quasi-stationary. Similar to stationary states, the dependence of the wave function of a particle on time is given in this case by the factor 
The complex quantity appears here as energy E, the imaginary part determines the probability of decay of a quasi-stationary state per unit time due to T. e.:
In quasiclassical When approaching, the probability given by f-loy (3) contains an exponential. factor of the same type as in-f-le (1). In the case of a spherically symmetric potential. barrier is the probability of decay of a quasi-stationary state from orbits. l determined by f-loy
Here r 1,2 are radial turning points, the integrand in which is equal to zero. Factor w 0 depends on the nature of the movement in the classically allowed part of the potential, for example. he is proportional. classic frequency of the particle between the barrier walls.
T. e. allows us to understand the mechanism of a-decay of heavy nuclei. Between the particle and the daughter nucleus there is an electrostatic force. repulsion determined by f-loy At small distances of the order of size A the nuclei are such that eff. potential can be considered negative: 
As a result, the probability A-decay is given by the relation
Here is the energy of the emitted a-particle.
T. e. determines the possibility of thermonuclear reactions occurring in the Sun and stars at temperatures of tens and hundreds of millions of degrees (see. Evolution of stars), as well as in terrestrial conditions in the form of thermonuclear explosions or CTS.
In a symmetric potential, consisting of two identical wells separated by a weakly permeable barrier, i.e. leads to states in wells, which leads to weak double splitting of discrete energy levels (so-called inversion splitting; see Molecular spectra). For an infinitely periodic set of holes in space, each level turns into a zone of energies. This is the mechanism for the formation of narrow electron energies. zones in crystals with strong coupling of electrons to lattice sites.
If an electric current is applied to a semiconductor crystal. field, then the zones of allowed electron energies become inclined in space. Thus, the post level electron energy crosses all zones. Under these conditions, the transition of an electron from one energy level becomes possible. zones to another due to T. e. The classically inaccessible area is the zone of forbidden energies. This phenomenon is called. Zener breakdown. Quasiclassical the approximation corresponds here to a small value of electrical intensity. fields. In this limit, the probability of a Zener breakdown is determined basically. exponential, in the cut indicator there is a large negative. a value proportional to the ratio of the width of the forbidden energy. zone to the energy gained by an electron in an applied field at a distance equal to the size of the unit cell.
A similar effect appears in tunnel diodes, in which the zones are inclined due to semiconductors R- And n-type on both sides of the border of their contact. Tunneling occurs due to the fact that in the zone where the carrier goes there is a finite density of unoccupied states.
Thanks to T. e. electric possible current between two metals separated by a thin dielectric. partition. These metals can be in both normal and superconducting states. In the latter case there may be Josephson effect.
T. e. Such phenomena occurring in strong electric currents are due. fields, such as autoionization of atoms (see Field ionization)And auto-electronic emissions from metals. In both cases, electric the field forms a barrier of finite transparency. The stronger the electric field, the more transparent the barrier and the stronger the electron current from the metal. Based on this principle scanning tunneling microscope- a device that measures the tunneling current from different points of the surface under study and provides information about the nature of its heterogeneity.
T. e. is possible not only in quantum systems consisting of a single particle. Thus, for example, low-temperature motion in crystals can be associated with tunneling of the final part of a dislocation, consisting of many particles. In problems of this kind, a linear dislocation can be represented as an elastic string, initially lying along the axis at in one of the local minima of the potential V(x, y). This potential does not depend on at, and its relief along the axis X is a sequence of local minima, each of which is lower than the other by an amount depending on the mechanical force applied to the crystal. . The movement of a dislocation under the influence of this stress is reduced to tunneling into an adjacent minimum defined. segment of a dislocation with subsequent pulling of its remaining part there. The same kind of tunnel mechanism may be responsible for the movement charge density waves in Peierls (see Peierls transition).
To calculate the tunneling effects of such multidimensional quantum systems, it is convenient to use semiclassical methods. representation of the wave function in the form 
Where S-classical system action. For T. e. the imaginary part is significant S, which determines the attenuation of the wave function in a classically inaccessible region. To calculate it, the method of complex trajectories is used.
Quantum particle overcoming potential. barrier may be connected to the thermostat. In classic Mechanically, this corresponds to motion with friction. Thus, to describe tunneling it is necessary to use a theory called dissipative. Considerations of this kind must be used to explain the finite lifetime of current states of Josephson contacts. In this case, tunneling occurs. quantum particle through the barrier, and the role of a thermostat is played by normal electrons.
Lit.: Landau L.D., Lifshits E.M., Quantum Mechanics, 4th ed., M., 1989; Ziman J., Principles of Solid State Theory, trans. from English, 2nd ed., M., 1974; Baz A. I., Zeldovich Ya. B., Perelomov A. M., Scattering, reactions and decays in nonrelativistic quantum mechanics, 2nd ed., M., 1971; Tunnel phenomena in solids, trans. from English, M., 1973; Likharev K.K., Introduction to the dynamics of Josephson junctions, M., 1985. B. I. Ivlev.
The tunnel effect is an amazing phenomenon, completely impossible from the standpoint of classical physics. But in the mysterious and mysterious quantum world, slightly different laws of interaction between matter and energy operate. The tunnel effect is the process of overcoming a certain potential barrier, provided that its energy is less than the height of the barrier. This phenomenon is exclusively quantum in nature and completely contradicts all the laws and dogmas of classical mechanics. The more amazing is the world in which we live.
The best way to understand what the quantum tunneling effect is is to use the example of a golf ball thrown into a hole with some force. At any given unit of time, the total energy of the ball is in opposition to the potential force of gravity. If we assume that it is inferior to the force of gravity, then the specified object will not be able to leave the hole on its own. But this is in accordance with the laws of classical physics. To overcome the edge of the hole and continue on its way, it will definitely need additional kinetic impulse. This is what the great Newton said.
In the quantum world, things are somewhat different. Now let’s assume that there is a quantum particle in the hole. In this case, we will no longer be talking about a real physical depression in the ground, but about what physicists conventionally call a “potential hole.” Such a value also has an analogue of the physical side - an energy barrier. Here the situation changes most radically. In order for the so-called quantum transition to take place and the particle to appear outside the barrier, another condition is necessary.
If the strength of the external energy field is less than the particle, then it has a real chance regardless of its height. Even if it does not have enough kinetic energy in the understanding of Newtonian physics. This is the same tunnel effect. It works as follows. It is typical to describe any particle not using any physical quantities, but through a wave function associated with the probability of the particle being located at a certain point in space at each specific unit of time.
When a particle collides with a certain barrier, using the Schrödinger equation, you can calculate the probability of overcoming this barrier. Since the barrier not only absorbs energy but also extinguishes it exponentially. In other words, in the quantum world there are no insurmountable barriers, but only additional conditions under which a particle can find itself beyond these barriers. Various obstacles, of course, interfere with the movement of particles, but are by no means solid, impenetrable boundaries. Conventionally speaking, this is a kind of borderland between two worlds - physical and energetic.
The tunnel effect has its analogue in nuclear physics - autoionization of an atom in a powerful electric field. Solid state physics also abounds in examples of tunneling manifestations. This includes field emission, migration, as well as effects that occur at the contact of two superconductors separated by a thin dielectric film. Tunneling plays an exceptional role in the implementation of numerous chemical processes under conditions of low and cryogenic temperatures.
Tunnel effect, tunneling- overcoming a potential barrier by a microparticle in the case when its total energy (which remains unchanged during tunneling) is less than the height of the barrier. The tunnel effect is an essentially natural phenomenon, impossible in; An analogue of the tunnel effect can be the penetration of a light wave into a reflecting medium (at distances of the order of the light wavelength) under conditions where, from the point of view, total internal reflection occurs. The phenomenon of tunneling underlies many important processes in molecular physics, in the physics of the atomic nucleus, etc.
Theory
The tunnel effect is ultimately explained by the relation (see also, Wave-particle duality). A classical particle cannot be inside a potential height barrier V, if its energy E< V, так как кинетическая энергия частицы p 2 / 2m = E − V becomes negative, and its momentum R- imaginary quantity ( m- particle mass). However, for a microparticle this conclusion is unfair: due to the uncertainty relationship, the fixation of a particle in the spatial region inside the barrier makes its momentum uncertain. Therefore, there is a non-zero probability of detecting a microparticle inside a region that is forbidden, from the point of view of classical mechanics. Accordingly, a certain probability of a particle passing through a potential barrier appears, which corresponds to the tunnel effect. This probability is greater, the smaller the mass of the particle, the narrower the potential barrier, and the less energy the particle lacks to reach the height of the barrier (that is, the smaller the difference V − E ).
The probability of passage through the barrier is the main factor determining the physical characteristics of the tunneling effect. In the case of a one-dimensional potential barrier, this characteristic is the barrier's transparency coefficient, equal to the ratio of the flux of particles passing through it to the flux incident on the barrier. In the case of a three-dimensional potential barrier limiting a closed region of space with reduced potential energy (potential well), the tunnel effect is characterized by the probability w exit of a particle from this region per unit time; magnitude w is equal to the product of the oscillation frequency of a particle inside a potential well and the probability of passing through the barrier. The possibility of “leakage” out of a particle that was initially located in a potential well leads to the fact that the corresponding particle energy levels acquire a finite width of the order of hw (h- ), and these states themselves become quasi-stationary.
Examples
An example of the manifestation of the tunnel effect in atomic physics is the processes of autoionization of an atom in a strong electric field. Recently, the process of ionization of an atom in the field of a strong electromagnetic wave has attracted especially much attention. In nuclear physics, the tunnel effect underlies the understanding of the laws of radioactive nuclei: as a result of the combined action of short-range nuclear attractive forces and electrostatic (Coulomb) repulsive forces, an alpha particle, when leaving the nucleus, has to overcome a three-dimensional potential barrier of the type described above (). Without tunneling, it would be impossible for thermonuclear reactions to occur: the barrier that prevents the convergence of reactant nuclei necessary for fusion is overcome partly due to the high speed (high temperature) of such nuclei, and partly due to the tunneling effect.
There are especially numerous examples of the manifestation of the tunnel effect in solid state physics: field emission of electrons from metals and semiconductors (see Tunnel emission); phenomena in semiconductors placed in a strong electric field (see); migration of valence electrons in the crystal lattice (see); effects that arise at the contact between two superconductors separated by a thin film of normal metal or dielectric (see), etc.
History and explorers
Literature
- Blokhintsev D.I., Fundamentals of Quantum Mechanics, 4th ed., M., 1963;
- Landau L. D., Lifshits E. M., Quantum mechanics. Nonrelativistic theory, 3rd ed., M., 1974 (Theoretical Physics, vol. 3).
TUNNEL EFFECT, a quantum effect consisting in the penetration of a quantum particle through a region of space, into which, according to the laws of classical physics, finding a particle is prohibited. Classic a particle with total energy E and in potential. field can only reside in those regions of space in which its total energy does not exceed the potential. energy U of interaction with the field. Since the wave function of a quantum particle is nonzero throughout space and the probability of finding a particle in a certain region of space is given by the square of the modulus of the wave function, then in forbidden (from the point of view of classical mechanics) regions the wave function is nonzero.
T It is convenient to illustrate the tunnel effect using a model problem of a one-dimensional particle in a potential field U(x) (x is the coordinate of the particle). In the case of a symmetrical double-well potential (Fig. a), the wave function must “fit” inside the wells, i.e., it is a standing wave. Discrete energy sources levels that are located below the barrier separating the minima of the potential form closely spaced (almost degenerate) levels. Energy difference levels, components, called. tunnel splitting, this difference is due to the fact that the exact solution of the problem (wave function) for each of the cases is localized in both minima of the potential and all exact solutions correspond to non-degenerate levels (see). The probability of the tunnel effect is determined by the coefficient of transmission of a wave packet through the barrier, which describes the non-stationary state of a particle localized in one of the potential minima.
Potential curves energy U (x) of a particle in the case when it is acted upon by an attractive force (a - two potential wells, b - one potential well), and in the case when a repulsive force acts on the particle (repulsive potential, c). E is the total energy of the particle, x is the coordinate. Thin lines depict wave functions.
In potential field with one local minimum (Fig. b) for a particle with energy E greater than the interaction potential at c =, discrete energy. there are no states, but there is a set of quasi-stationary states, in which the great relates. the probability of finding a particle near the minimum. Wave packets corresponding to such quasi-stationary states describe metastable ones; wave packets spread out and disappear due to the tunnel effect. These states are characterized by their lifetime (probability of decay) and energy width. level.
For a particle in a repulsive potential (Fig. c), a wave packet describing a non-stationary state on one side of the potential. barrier, even if the energy of a particle in this state is less than the height of the barrier, it can, with a certain probability (called the probability of penetration or the probability of tunneling), pass on the other side of the barrier.
Naib. important for the manifestation of the tunnel effect: 1) tunnel splitting of discrete oscillations, rotation. and electronic-co-lebat. levels. Splitting of oscillations. levels in with several. equivalent equilibrium nuclear configurations is inversion doubling (in type), splitting of levels in with inhibited internal. rotation ( , ) or in , for which intra-mol. rearrangements leading to equivalent equilibrium configurations (eg PF 5). If different equivalent minima are not separated by potential. barriers (for example, equilibrium configurations for right- and left-handed complexes), then an adequate description of real piers. systems is achieved using localized wave packets. In this case, stationary states localized in two minima are unstable: under the influence of very small perturbations, the formation of two states localized in one or another minimum is possible.
The splitting of quasi-degenerate groups rotates. states (so-called rotational clusters) is also due to tunneling of the mol. systems between several neighborhoods. equivalent stationary axes of rotation. Splitting of electron vibrations. (vibronic) states occurs in the case of strong Jahn-Teller effects. Tunnel splitting is also associated with the existence of bands formed by electronic states of individual or molecular states. fragments in periodic structure.
2) Phenomena of particle transfer and elementary excitations. This set of phenomena includes non-stationary processes that describe transitions between discrete states and the decay of quasi-stationary states. Transitions between discrete states with wave functions localized in different states. minimums of one adiabatic. potential, correspond to a variety of chemicals. r-tions. The tunnel effect always makes a certain contribution to the rate of transformation, but this contribution is significant only at low temperatures, when the above-barrier transition from the initial state to the final state is unlikely due to the low population of the corresponding energy levels. The tunnel effect manifests itself in the non-Arrhenius behavior of the r-tion velocity; A typical example is the growth of a chain during radiation-initiated solids. The speed of this process at temperature is approx. 140 K is satisfactorily described by the Arrhenius law with