Among the many perplexing results of quantum mechanics is one that contradicts a result from introductory physics: the possibility of finding a quantum particle in a region that would be forbidden classically by energy conservation. An especially interesting example of this phenomenon with practical applications is quantum tunneling. Here we investigate the reasons for this puzzling result by focusing on the difference between how quantities like kinetic and potential energy are represented mathematically in classical and quantum mechanics. In quantum mechanics, physical observables, like energy, are represented by operators rather than real numbers. The consequences of this difference will be illustrated explicitly using a toy model in which the kinetic and potential energy operators are represented by $2 \times 2$ matrices, which do not commute like their classical analogs. This model will then illustrate how classically perplexing results, like a quantum particle being found in the forbidden region, can arise.
Among the many perplexing results of quantum mechanics is one that contradicts a result from introductory physics: the possibility of finding a quantum particle in a region that would be forbidden classically by energy conservation. An especially interesting example of this phenomenon with practical applications is quantum tunneling. Here we investigate the reasons for this puzzling result by focusing on the difference between how quantities like kinetic and potential energy are represented mathematically in classical and quantum mechanics. In quantum mechanics, physical observables, like energy, are represented by operators rather than real numbers. The consequences of this difference will be illustrated explicitly using a toy model in which the kinetic and potential energy operators are represented by 2×2 matrices, which do not commute like their classical analogs. This model will then illustrate how classically perplexing results, like a quantum particle being found in the forbidden region, can arise.
Classical mechanics is a singular theory in that real-energy classical particles can never enter classically forbidden regions. However, if one regulates classical mechanics by allowing the energy E of a particle to be complex, the particle exhibits quantum-like behavior: complex-energy classical particles can travel between classically allowed regions separated by potential barriers. When Im E → 0, the classical tunneling probabilities persist. Hence, one can interpret quantum tunneling as an anomaly. A numerical comparison of complex classical tunneling probabilities with quantum tunneling probabilities leads to the conjecture that as Re E increases, complex classical tunneling probabilities approach the corresponding quantum probabilities. Thus, this work attempts to generalize the Bohr correspondence principle from classically allowed to classically forbidden regions.
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.
Classical mechanics and quantum mechanics contradict each other, and both are essential to explain the phenomena of the exclusively different realm of nature. On the other hand, Bohr's correspondence principle shows classical mechanics is the somewhat approximate version of quantum mechanics. Classically a particle with negative energy i. e. V E < is not allowed go through a forbidden region or disappearing from one well to another well. This paper gives the numerical studies for the trajectory of the particle in a double-well potential and presents quantum mechanical behavior such as tunneling in the complex plane for different energies. Our findings provide a route to solve the classical system with complex energy.
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).
Introduction In classical physics it is meaningful to ask the question, ‘How much time does a particle take to pass through a given region?’ The interesting question in quantum mechanics is: does a particle take a definite time to tunnel through a classically forbidden region? The question has been debated ever since the idea that there was such a time in quantum theory was first put forward by MacColl way back in 1932 [253]. A plethora of times has since then been proposed, and the answer seems to depend on the interpretation of quantum mechanics one uses. A reliable answer is clearly of great importance for the design of high-frequency quantum devices, tunnelling phenomena (as, for example, in scanning tunneling microscopy), nuclear and chemical reactions and, of course, for purely conceptual reasons. Most of the controversies centre around simple and intuitive notions in idealized one-dimensional models in a scattering configuration in which a particle (usually represented by a wave packet) is incident on a potential barrier localized in the interval [ a, b ]. Three kinds of time have been defined in this context. One, called the transmission time τ T ( a, b ), is the average time spent within the barrier region by the particles that are eventually transmitted. Similarly, the reflection time τ R ( a, b ) is the average time spent within the barrier region by the particles that are eventually reflected.
This paper proposes a classical alternative explanation for the tunneling effect in quantum mechanics. Standard quantum theory describes tunneling as a particle "penetrating" a potential barrier whose energy exceeds the particle's own kinetic energy with a certain probability. However, this explanation focuses only on the mathematical initial and final states, completely ignoring any dynamical process the electron may undergo before the barrier, and lacks a physical picture. This paper argues that tunneling is not an intrinsic quantum property of a single electron, but rather a collective statistical fluctuation event involving multiple electrons: electrons near the barrier, through a series of random collisions in the correct direction, gradually transfer and concentrate kinetic energy onto the electron closest to the barrier, enabling it to instantaneously acquire kinetic energy exceeding the barrier threshold and thus classically fly over the barrier. During the crossing, the electron must do work against the barrier, converting kinetic energy into potential energy; therefore, after crossing, its kinetic energy is reduced and it cannot cross the same barrier again. Tunneling probability is reinterpreted as the success probability of such a "directed energy relay" event. This model is fully self-consistent within the framework of classical electromagnetism and statistical mechanics, requiring no wave functions or probability waves, with a clear physical picture and explicit causal chain. This paper further points out that under strictly single-electron conditions, this model predicts that tunneling will completely disappear, a prediction that stands in fundamental opposition to the standard quantum model and can serve as a decisive experiment to test the two theories.