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 represents 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 -total energy particles, x - 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 the particle in this state less height barrier, can with a certain probability (called penetration probability or tunneling probability) pass through 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, the case is calibrated in two minima stationary states 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 zones formed electronic states individual or mol. 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 speed of movement, but this contribution is significant only when low t-rah, when an over-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; typical example- chain growth during radiation-initiated solid. The speed of this process at temperature is approx. 140 K is satisfactorily described by the Arrhenius law with
- Translation
I'll start with two simple questions with fairly intuitive answers. Let's take a bowl and a ball (Fig. 1). If I need to:
The ball remained motionless after I placed it in the bowl, and
it remained in approximately the same position when moving the bowl,
So where should I put it?
Rice. 1
Of course, I need to put it in the center, at the very bottom. Why? Intuitively, if I put it somewhere else, it will roll to the bottom and flop back and forth. As a result, friction will reduce the height of the dangling and slow it down below.
In principle, you can try to balance the ball on the edge of the bowl. But if I shake it a little, the ball will lose its balance and fall. So this place doesn't meet the second criterion in my question.
Let us call the position in which the ball remains motionless, and from which it does not deviate much with small movements of the bowl or ball, “stable position of the ball.” The bottom of the bowl is such a stable position.
Another question. If I have two bowls like in fig. 2, where will be the stable positions for the ball? This is also simple: there are two such places, namely, at the bottom of each of the bowls.
Rice. 2
Finally, another question with an intuitive answer. If I place a ball at the bottom of bowl 1, and then leave the room, close it, ensure that no one goes in there, check that there have been no earthquakes or other shocks in this place, then what are the chances that in ten years when I If I open the room again, I will find a ball at the bottom of bowl 2? Of course, zero. In order for the ball to move from the bottom of bowl 1 to the bottom of bowl 2, someone or something must take the ball and move it from place to place, over the edge of bowl 1, towards bowl 2 and then over the edge of bowl 2. Obviously, the ball will remain at the bottom of the bowl 1.
Obviously and essentially true. And yet, in quantum world, in which we live, no object remains truly motionless, and its position is precisely unknown. So none of these answers are 100% correct.
Tunneling
Rice. 3
If I place an elementary particle like an electron in a magnetic trap (Fig. 3) that works like a bowl, tending to push the electron towards the center in the same way that gravity and the walls of the bowl push the ball towards the center of the bowl in Fig. 1, then what will be the stable position of the electron? As one would intuitively expect, the average position of the electron will be stationary only if it is placed at the center of the trap.
But quantum mechanics adds one nuance. The electron cannot remain stationary; its position is subject to "quantum jitter". Because of this, its position and movement are constantly changing, or even have a certain amount of uncertainty (this is the famous “uncertainty principle”). Only the average position of the electron is at the center of the trap; if you look at the electron, it will be somewhere else in the trap, close to the center, but not quite there. An electron is stationary only in this sense: it usually moves, but its movement is random, and since it is trapped, on average it does not move anywhere.
This is a little strange, but it just reflects the fact that an electron is not what you think it is and does not behave like any object you have seen.
This, by the way, also ensures that the electron cannot be balanced at the edge of the trap, unlike the ball at the edge of the bowl (as below in Fig. 1). The position of the electron is not precisely defined, so it cannot be precisely balanced; therefore, even without shaking the trap, the electron will lose its balance and fall off almost immediately.
But what's weirder is the case where I'll have two traps separated from each other, and I'll place an electron in one of them. Yes, the center of one of the traps is a good, stable position for the electron. This is true in the sense that the electron can remain there and will not escape if the trap is shaken.
However, if I place an electron in trap No. 1 and leave, close the room, etc., there is a certain probability (Fig. 4) that when I return the electron will be in trap No. 2.
Rice. 4
How did he do it? If you imagine electrons as balls, you won't understand this. But electrons are not like marbles (or at least not like your intuitive idea of marbles), and their quantum jitter gives them an extremely small but non-zero chance of "walking through walls" - the seemingly impossible possibility of moving to the other side. This is called tunneling - but don't think of the electron as digging a hole in the wall. And you will never be able to catch him in the wall - red-handed, so to speak. It's just that the wall isn't completely impenetrable to things like electrons; electrons cannot be trapped so easily.
In fact, it's even crazier: since it's true for an electron, it's also true for a ball in a vase. The ball may end up in vase 2 if you wait long enough. But the likelihood of this is extremely low. So small that even if you wait a billion years, or even billions of billions of billions of years, it won’t be enough. From a practical point of view, this will “never” happen.
Our world is quantum, and all objects consist of elementary particles and obey the rules of quantum physics. Quantum jitter is always present. But most of objects whose mass is large compared to the mass of elementary particles - a ball, for example, or even a speck of dust - this quantum jitter is too small to be detected, except in specially designed experiments. And the resulting possibility of tunneling through walls is also not observed in ordinary life.
In other words: any object can tunnel through a wall, but the likelihood of this usually decreases sharply if:
At the object large mass,
the wall is thick (large distance between two sides),
the wall is difficult to overcome (it takes a lot of energy to break through a wall).
In principle the ball can get over the edge of the bowl, but in practice this may not be possible. It can be easy for an electron to escape from a trap if the traps are close and not very deep, but it can be very difficult if they are far away and very deep.
Is tunneling really happening?
Rice. 5
Or maybe this tunneling is just a theory? Absolutely not. It is fundamental to chemistry, occurs in many materials, plays a role in biology, and is the principle used in our most sophisticated and powerful microscopes.
For the sake of brevity, let me focus on the microscope. In Fig. Figure 5 shows an image of atoms taken using a scanning tunneling microscope. This microscope has a narrow needle whose tip moves in close proximity to the material being studied (see Fig. 6). The material and the needle are, of course, made of atoms; and at the back of the atoms are electrons. Roughly speaking, electrons are trapped inside the material being studied or at the tip of the microscope. But the closer the tip is to the surface, the more likely the tunneling transition of electrons between them is. A simple device (a potential difference is maintained between the material and the needle) ensures that electrons will prefer to jump from the surface to the needle, and this flow - electricity, measurable. The needle moves over the surface, and the surface appears closer or further from the tip, and the current changes - it becomes stronger as the distance decreases and weaker as it increases. By monitoring the current (or, conversely, moving the needle up and down to maintain direct current) when scanning a surface, the microscope makes a conclusion about the shape of this surface, and often the detail is enough to make out individual atoms.
Rice. 6
Tunneling plays many other roles in nature and modern technologies.
Tunneling between traps of different depths
In Fig. 4 I meant that both traps had the same depth - just like both bowls in fig. 2 same shape. This means that an electron, being in any of the traps, is equally likely to jump to the other.Now let us assume that one electron trap in Fig. 4 deeper than the other - exactly the same as if one bowl in fig. 2 was deeper than the other (see Fig. 7). Although an electron can tunnel in any direction, it will be much easier for it to tunnel from a shallower to a deeper trap than vice versa. Accordingly, if we wait long enough for the electron to have enough time to tunnel in either direction and return, and then start taking measurements to determine its location, we will most often find it deeply trapped. (In fact, there are some nuances here too; everything also depends on the shape of the trap). Moreover, the difference in depth does not have to be large for tunneling from a deeper to a shallower trap to become extremely rare.
In short, tunneling will generally occur in both directions, but the probability of going from a shallow to a deep trap is much greater.
Rice. 7
It is this feature that a scanning tunneling microscope uses to ensure that electrons only travel in one direction. Essentially, the tip of the microscope needle is trapped deeper than the surface being studied, so electrons prefer to tunnel from the surface to the needle rather than vice versa. But the microscope will work in the opposite case. The traps are made deeper or shallower by using a power source that creates a potential difference between the tip and the surface, which creates a difference in energy between the electrons on the tip and the electrons on the surface. Since it is quite easy to make electrons tunnel more often in one direction than another, this tunneling becomes practically useful for use in electronics.
1.7. The concept of the tunnel effect.
The tunnel effect is the passage of particles through potential barrier due to wave properties particles.
Let a particle moving from left to right encounter a potential barrier of height U 0 and width l. According to classical concepts, a particle passes unhindered over a barrier if its energy E greater than the barrier height ( E> U 0 ). If the particle energy is less than the barrier height ( E< U 0 ), then the particle is reflected from the barrier and begins to move in the opposite direction; the particle cannot penetrate through the barrier.
Quantum mechanics takes into account the wave properties of particles. For a wave, the left wall of the barrier is the boundary of two media, at which the wave is divided into two waves - reflected and refracted. Therefore, even with E> U 0 it is possible (albeit with a small probability) that a particle is reflected from the barrier, and when E< U 0 there is a nonzero probability that the particle will be on the other side of the potential barrier. In this case, the particle seemed to “pass through a tunnel.”
Let's decide the problem of a particle passing through a potential barrier for the simplest case of a one-dimensional rectangular barrier, shown in Fig. 1.6. The shape of the barrier is specified by the function
. (1.7.1)
Let us write the Schrödinger equation for each of the regions: 1( x<0 ), 2(0< x< l) and 3( x> l):
;
(1.7.2)
; (1.7.3)
. (1.7.4)
Let's denote
(1.7.5)
. (1.7.6)
General solutions of equations (1), (2), (3) for each of the areas have the form:
Solution of the form 
corresponds to a wave propagating in the direction of the axis x, A 
- a wave propagating in the opposite direction. In region 1 term 
describes a wave incident on a barrier, and the term 
- wave reflected from the barrier. In region 3 (to the right of the barrier) there is only a wave propagating in the x direction, so 
.
The wave function must satisfy the continuity condition, therefore solutions (6), (7), (8) at the boundaries of the potential barrier must be “stitched”. To do this, we equate the wave functions and their derivatives at x=0 And x = l:
;
;
;
. (1.7.10)
Using (1.7.7) - (1.7.10), we obtain four equations to determine five coefficients A 1 , A 2 , A 3 ,IN 1 And IN 2 :
A 1 +B 1 =A 2 +B 2 ;
A 2 exp( l) + B 2 exp(- l)= A 3 exp(ikl) ;
ik(A 1 - IN 1 ) = (A 2 -IN 2 ) ; (1.7.11)
(A 2 exp( l)-IN 2 exp(- l) = ikA 3 exp(ikl) .
To obtain the fifth relation, we introduce the concepts of reflection coefficients and barrier transparency.
Reflection coefficient let's call the relation
, (1.7.12)
which defines probability reflection of a particle from a barrier.
Transparency factor
(1.7.13)
gives the probability that the particle will pass through the barrier. Since the particle will either be reflected or pass through the barrier, the sum of these probabilities is equal to one. Then
R+ D =1; (1.7.14)
. (1.7.15)
That's what it is fifth relationship that closes the system (1.7.11), from which all five coefficients
Of greatest interest is transparency coefficientD. After transformations we get
,
(7.1.16)
Where D 0 – value close to unity.
From (1.7.16) it is clear that the transparency of the barrier strongly depends on its width l, on how high the barrier is U 0 exceeds the particle energy E, and also on the mass of the particle m.
WITH 
from the classical point of view, the passage of a particle through a potential barrier at E<
U 0
contradicts the law of conservation of energy. The fact is that if a classical particle were at some point in the barrier region (region 2 in Fig. 1.7), then its total energy would be less than the potential energy (and the kinetic energy would be negative!?). WITH quantum dot there is no such contradiction. If a particle moves towards a barrier, then before colliding with it it has a very specific energy. Let the interaction with the barrier last for a while
t, then, according to the uncertainty relation, the energy of the particle will no longer be definite; energy uncertainty 
. When this uncertainty turns out to be on the order of the height of the barrier, it ceases to be an insurmountable obstacle for the particle, and the particle will pass through it.
The transparency of the barrier decreases sharply with its width (see Table 1.1.). Therefore, particles can pass through only very narrow potential barriers due to the tunneling mechanism.
Table 1.1
Values of the transparency coefficient for an electron at ( U 0 – E ) = 5 eV = const
|
l, nm | |||||
We considered a rectangular shaped barrier. In the case of a potential barrier of arbitrary shape, for example, as shown in Fig. 1.7, the transparency coefficient has the form
. (1.7.17)
The tunnel effect manifests itself in a number of physical phenomena and has important practical applications. Let's give some examples.
1. Field electron (cold) emission of electrons.
IN 
In 1922, the phenomenon of cold electron emission from metals under the influence of a strong external electric field was discovered. Potential Energy Graph U electron from coordinate x shown in Fig. At x
<
0 is the region of the metal in which electrons can move almost freely. Here the potential energy can be considered constant. A potential wall appears at the metal boundary, preventing the electron from leaving the metal; it can do this only by acquiring additional energy, equal to work exit A.
Outside the metal (at x >
0) the energy of free electrons does not change, so when x> 0 the graph U(x)
goes horizontally. Let us now create a strong electric field near the metal. To do this, take a metal sample in the shape of a sharp needle and connect it to the negative pole of the source. Rice. 1.9 Operating principle of a tunnel microscope
ka voltage, (it will be the cathode); We will place another electrode (anode) nearby, to which we will connect the positive pole of the source. If the potential difference between the anode and the cathode is large enough, it is possible to create an electric field with a strength of about 10 8 V/m near the cathode. The potential barrier at the metal-vacuum interface becomes narrow, electrons leak through it and leave the metal.
Field emission was used to create vacuum tubes with cold cathodes (they are now practically out of use); it has now found application in tunnel microscopes, invented in 1985 by J. Binning, G. Rohrer and E. Ruska.
In a tunnel microscope, a probe - a thin needle - moves along the surface under study. The needle scans the surface under study, being so close to it that electrons from the electron shells (electron clouds) of surface atoms, due to wave properties, can reach the needle. To do this, we apply a “plus” from the source to the needle, and a “minus” to the sample under study. The tunnel current is proportional to the transparency coefficient of the potential barrier between the needle and the surface, which, according to formula (1.7.16), depends on the barrier width l. When scanning the surface of a sample with a needle, the tunneling current varies depending on the distance l, repeating the surface profile. Precision movements of the needle over short distances are carried out using the piezoelectric effect; for this, the needle is fixed on a quartz plate, which expands or contracts when an electrical voltage is applied to it. Modern technologies make it possible to produce a needle so thin that there is only one atom at its end.
AND 
the image is formed on the computer display screen. Permission tunnel microscope so high that it allows you to “see” the arrangement of individual atoms. Figure 1.10 shows an example image of the atomic surface of silicon.
2. Alpha radioactivity ( – decay). In this phenomenon, a spontaneous transformation of radioactive nuclei occurs, as a result of which one nucleus (it is called the mother nucleus) emits an particle and turns into a new (daughter) nucleus with a charge less than 2 units. Let us recall that the particle (the nucleus of a helium atom) consists of two protons and two neutrons.
E 
If we assume that the α-particle exists as a single formation inside the nucleus, then the graph of the dependence of its potential energy on the coordinate in the field of the radioactive nucleus has the form shown in Fig. 1.11. It is determined by the energy of the strong (nuclear) interaction, caused by the attraction of nucleons to each other, and the energy of the Coulomb interaction (electrostatic repulsion of protons).
As a result, is a particle in the nucleus with energy E is located behind the potential barrier. Due to its wave properties, there is some probability that the particle will end up outside the nucleus.
3. Tunnel effect inp- n- transition used in two classes of semiconductor devices: tunnel And reversed diodes. A feature of tunnel diodes is the presence of a falling section on the direct branch of the current-voltage characteristic - a section with a negative differential resistance. The most interesting thing about reverse diodes is that when turned in reverse, the resistance is less than when turned in reverse. For more information on tunnel and reverse diodes, see section 5.6.
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 its potential energy 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 potential energy will be 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 Newtonian world ball and hole, the very fact that, having gone over the side of the hole, the ball will roll further, does not make sense if the ball does not have enough kinetic energy to reach the upper 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 what height above the surface outside the edge of the side is.
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're talking about no longer about a real physical pit, but about 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 tension energy field below , than the energy that a particle possesses, 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’s location in this place V this moment time. If a particle collides with a potential barrier, Schrödinger's equation allows one to calculate the probability of a 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, a potential barrier in the world quantum mechanics blurry It, of course, prevents the movement of the particle, but is not a solid, impenetrable boundary, as is the case in classical mechanics Newton.
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 get experimental confirmation reality of the tunnel effect.
Another important example tunnel effect - the process of thermonuclear fusion that supplies energy to stars ( cm. 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 equal charge(V in this case protons that make up deuterium nuclei) acts most powerful force 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 interior of stars, the temperature and pressure are so high that the energy of the nuclei approaches the threshold of their synthesis (in our sense, the nuclei are almost at the edge of the barrier), as a result of which the tunnel effect, happens thermonuclear fusion- 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 to the maximum close range along the surface being examined, he sorts it out atom by atom. When the probe is in close proximity to atoms, the barrier is lower , than when the probe passes in the spaces 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 detailed investigation atomic structures 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 effect
Tunnel effect (tunneling) – the passage of a particle (or system) through a region of space in which stay is prohibited classical mechanics. Most famous example such a process is the passage of a particle through a potential barrier when its energy E is less than the barrier height U 0 . In classical physics, a particle cannot appear in the region of such a barrier, much less pass through it, since this violates the law of conservation of energy. However, in quantum physics the situation is fundamentally different. Quantum particle does not move along any specific trajectory. Therefore, we can only talk about the probability of finding a particle in a certain region of space ΔрΔх > ћ. In this case, neither potential nor kinetic energies have definite values in accordance with the uncertainty principle. A deviation from the classical energy E by the amount ΔE is allowed during time intervals t given by the uncertainty relation ΔEΔt > ћ (ћ = h/2π, where h is Planck’s constant).
The possibility of a particle passing through a potential barrier is due to the requirement of continuous wave function on the walls of the potential barrier. The probability of detecting a particle on the right and left is related to each other by a relationship that depends on the difference E - U(x) in the region of the potential barrier and on the barrier width x 1 - x 2 at a given energy.
As the height and width of the barrier increases, the probability of a tunnel effect decreases exponentially. The probability of a tunnel effect also decreases rapidly with increasing particle mass.
Penetration through the barrier is probabilistic. Particle with E< U 0 , натолкнувшись на барьер, может либо пройти сквозь него,
либо отразиться. Суммарная вероятность этих двух возможностей равна 1. Если
на барьер падает поток частиц с Е < U 0 , то часть этого
потока будет просачиваться сквозь барьер, а часть – отражаться. Туннельное
прохождение частицы через потенциальный барьер лежит в основе многих явлений
ядерной и atomic physics: alpha decay, cold emission of electrons from metals, phenomena in the contact layer of two semiconductors, etc.