A tennis ball is not expected to penetrate through a brick wall since a motion under a barrier is impossible in classical mechanics. With quantum effects a motion of a particle through a barrier is allowed due to quantum tunneling. According to usual theories of tunneling, the particle density decays inside a classical barrier resulting in an extremely slow pentration process. However, there are no general laws forbidding fast motion through classical barriers. The problem addressed is investigation of unusual features o quantum tunneling through a classic static barrier which is at least two-dimensional. Here we show that penetration through such barrier can be not slow. When the barrier satisfies the certain conditions, a regime of quantum lens is possible with formation of caustics. De Broglie waves are reflected from the caustics, interfere, and result in a not small flux from under the barrier. This strongly contrasts to the usual scenario with a decaying under-barrier density. We construct a particular example of fast motion through a classical barrier. One can unexectedly conclude that, in principle, nature allows fast penetration through classical barriers which against common sense. The phenomenon may be responsible for a variety of processes in labs and nature. For example, tunneling in solids may occur with a different scenario, in biophysics and chemistry one can specify conditions for unusual reactions, and evanescent optical waves may strongly change their properties. In condensed matter and cosmic physics there are phenomena with misterious reasons of an energy emission, for instance, gamma-ray bursts. One can try to treat them in the context of fast escape from under some barriers.
A tennis ball is not expected to penetrate through a brick wall since a motion under a barrier is impossible in classical mechanics. With quantum effects a motion of a particle through a barrier is allowed due to quantum tunneling. According to usual theories of tunneling, the particle density decays inside a classical barrier resulting in an extremely slow pentration process. However, there are no general laws forbidding fast motion through classical barriers. The problem addressed is investigation of unusual features o quantum tunneling through a classic static barrier which is at least two-dimensional. Here we show that penetration through such barrier can be not slow. When the barrier satisfies the certain conditions, a regime of quantum lens is possible with formation of caustics. De Broglie waves are reflected from the caustics, interfere, and result in a not small flux from under the barrier. This strongly contrasts to the usual scenario with a decaying under-barrier density. We construct a particular example of fast motion through a classical barrier. One can unexectedly conclude that, in principle, nature allows fast penetration through classical barriers which against common sense. The phenomenon may be responsible for a variety of processes in labs and nature. For example, tunneling in solids may occur with a different scenario, in biophysics and chemistry one can specify conditions for unusual reactions, and evanescent optical waves may strongly change their properties. In condensed matter and cosmic physics there are phenomena with misterious reasons of an energy emission, for instance, gamma-ray bursts. One can try to treat them in the context of fast escape from under some barriers.
Abstract A distinguishing feature of quantum mechanics, as compared with classical mechanics, is the phenomenon of matter exhibiting wave-like properties. A direct consequence of this wave property of particles is the tunneling of particles through potential energy barriers.-3 The classical particle is forbidden from penetrating the barrier region. In this chapter, the problem of one-dimensional (1D) quantum mechanical tunneling of particles through potential barriers of arbitrary shape is treated. We show that tunneling is a mixed boundary value problem, and obtain the action integral that leads to Schrodinger’s equation and at the same time gives the equations defining the boundary conditions (BCs).
It is possible in quantum mechanics to sneak quickly across a region which is illegal energetically. Richard Feynman Barrier penetration One of the most startling consequences of de Broglie's wave hypothesis and Schrödinger's equation was the discovery that quantum objects can ‘tunnel’ through potential energy barriers that classical particles are forbidden to penetrate. To gain some idea of what we mean by an energy barrier, let us go back to our roller coaster and look at a larger section of track, as shown in Fig. 5.1. If we start the carriage from rest, high up on the left, at A, and ignore any small frictional energy losses, we know from the conservation of energy that we shall arrive on the other side at the same height we started from, at C. As we went over the little hill B, at the bottom of the valley, the car slowed down as some of our kinetic energy was changed to potential energy in climbing the hill, but because we started much higher up, we had plenty of energy to spare to get us over the top. However, if we started the carriage from rest at A, we do not have enough energy to climb over the hill D and get to E. This is an example of an ‘energy barrier’, and we can say that the region from C to E is ‘classically forbidden’. What is remarkable about quantum ‘particles’ is that they do not behave like these classical objects.
Since the early days of quantum mechanics, tunneling has been recognized as one of the hallmarks of the wave character of microscopic physics. The possibility of a quantum particle to penetrate an energetic barrier represents certainly one of the most spectacular implications of quantum theory and has led to various applications in nuclear, atomic, and molecular physics as well as in mesoscopic science. Typical scenarios in which tunneling manifests are the escape route of a quantum particle from a quasibounded region, the transition between two or more symmetry-related, but classically disconnected wells (which we shall focus on in the following), as well as scattering or transport through potential barriers. The spectrum of scenarios becomes even richer when the concept of tunneling is generalized to any kind of classically forbidden transitions in phase space, that is, to transitions that are not necessarily inhibited by static potential barriers but by some other constraints of the underlying classical dynamics (such as integrals of motion). Such “dynamical tunneling” processes arise frequently in molecular systems [1] and were realized with cold atoms propagating in periodically modulated optical lattices [2-4]. Moreover, the electromagnetic analog of dynamical tunneling was also obtained with microwaves in billiards [5].
Quantum tunnelling, a hallmark phenomenon of quantum mechanics, allows particles to pass through the classically forbidden region. It underpins fundamental processes ranging from nuclear fusion and photosynthesis to the operation of superconducting qubits. Yet the underlying dynamics of particle motion during tunnelling remain subtle and are still the subject of active debate. Here, by analyzing the temporal evolution of the tunnelling process, we show that the particle velocity inside the barrier continuously relaxes from a large initial value toward a smaller one, and may even approach zero in the evanescent regime. Meanwhile, the probability density within the barrier gradually builds up before reaching its stationary profile, in contrast to existing inherently. In addition, starting from the steady-state equations, we derive an explicit relation between the particle velocity and the barrier width, and show that the velocity in evanescent states approaches zero when the barrier is sufficiently wide. These findings resolve the apparent paradox of a vanishing steady-state velocity coexisting with a finite particle density. We point out that defining an effective speed from the probability density, rather than from the probability current, can lead to spuriously nonzero "stationary speed," as appears to be the case in Ref. [Nature 643, 67 (2025)]. Our work establishes a clear dynamical picture for the formation of tunnelling flow and provides a theoretical foundation for testing time-resolved tunnelling phenomena.
When we dealt with the finite potential well in the last chapter, it was observed that the wave function penetrated into the barriers. This process does not occur in classical mechanics, since a particle will actually bounce off the barrier. However, when we treat the particle as a wave, then the wave nature of barrier penetration can occur. This is familiar in electromagnetic waves, where the decaying wave (as opposed to a propagating wave) is termed an evanescent wave. For energies below the top of the barrier, the wave is attenuated, and it decays exponentially. Yet, it takes a significant distance for this decay to eliminate the wave completely. If the barrier is thinner than this critical distance, the evanescent wave can excite a propagating wave in the region beyond the barrier. In optics, this effect is termed frustrated total internal reflection . We are familiar with total internal reflection just from a swimming pool. Water has a higher dielectric constant than free space, so when we are under water, there is critical angle beyond which we can no longer see out (from under the water) of the pool. In optics, we can use a prism, and there will be reflection from the back surface for angles greater than a critical angle. The wave in back of the prism is attenuated and decaying. But, if we place another high dielectric constant material close to the prism, the decaying wave can excite a wave in this new material. 1 Thus, the wave can penetrate the barrier and continue to propagate, with an attenuated amplitude, in the trans-barrier region. For particles, this process is termed tunneling , with analogy to the miners who burrow through a mountain in order to get to the other side! In fact, it is known that the nuclear fusion process by which our sun produces its power would not occur were there no tunneling. The intra-nuclear repulsion is just too strong and tunneling is required for the nuclei to find one another.
At the turn of the twentieth century, the establishment of quantum theory propelled rapid advancements, particularly in the understanding of quantum tunnelling—a fundamental phenomenon in quantum mechanics crucial for various physical processes. The quantum phenomenon of particle passes through potential barriers is of great importance. In classical physics, when the energy of a particle is less than the height of a double barrier structure, it is impossible for it to pass through. However, quantum mechanics allows a particle to penetrate the barrier and emerge on the other side. This paper explores the quantum tunnelling effect, focusing on the single potential barrier model in one dimension and subsequently extending to the double potential barrier model. The Schrödinger equation provides the foundational framework for elucidating the motion of microscopic particles, emphasizing wave-particle duality inherent in quantum mechanics. The analysis of the single potential barrier model involves solving the Schrödinger equation in different regions, determining wave functions and coefficients through boundary conditions. The transmission coefficient is derived, representing the probability of a particle passing through a barrier. In the case of a “thick” barrier, an approximate form for transmission coefficient is provided, demonstrating the exponential decrease in transmission probability with increasing barrier thickness.