L. S. F. Olavo
In this continuation paper we will address the problem of tunneling. We will show how to settle this phenomenon within our classical interpretation. It will be shown that, rigorously speaking, there is no tunnel effect at all.
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The results, different aspects and applications of our method of quantisation on configuration manifolds - called Borel Quantisation - were presented at meetings of the series `Symmetries in Science' and can be found in the published proceedings. The developments with numerous coauthors, on Borel quantisation and the related family of nonlinear Schrödinger equations called Doebner-Goldin equations, are reviewed and commented here.
We calculate a tunneling time distribution by means of Nelson's quantum mechanics and investigate its statistical properties. The relationship between the average and deviation of tunneling time suggests the exsistence of ``wave-particle duality'' in the tunneling phenomena.
We calculate the time taken by a wave packet to travel through a classically forbidden region of space in space fractional quantum mechanics. We obtain the close form expression of tunneling time from a rectangular barrier by stationary phase method. We show that tunneling time depends upon the width $b$ of the barrier for $b \to \infty$ and therefore Hartman effect doesn't exist in space fractional quantum mechanics. Interestingly we found that the tunneling time monotonically reduces with increasing $b$. The tunneling time is smaller in space fractional quantum mechanics as compared to the c…
We analyze the effects of inelastic scattering on the tunneling time theoretically, using generalized Nelson's quantum mechanics. This generalization enables us to describe quantum system with optical potential and channel couplings in a real time stochastic approach, which seems to give us a new insight into quantum mechanics beyond Copenhagen interpretation.
Recent developments in elementary quantum mechanics have seen a number of extraordinary claims regarding quantum behaviour, and even questioning internal consistency of the theory. These are, we argue, different disguises of what Feynman described as quantum theory's "only mystery".
Recently, people have caculated tunneling's characteristic times within Bohmian mechanics. Contrary to some characteristic times defined within the framework of the standard interpretation of quantum mechanics, these have reasonable values. Here, we introduce one of available definitions for tunnelling's characteristic times within the standard interpretation as the best definition that can be accepted for the tunneling times. We show that, due to experimental limitations, Bohmian mechanics leads to same tunneling times.
We calculate the time taken by a wave packet to travel through a classically forbidden locally periodic rectangular potential in space fractional quantum mechanics (SFQM). We obtain the close form expression of tunneling time from such a potential by stationary phase method. We show that tunneling time depends upon the width $b$ of the single barrier and separation $L$ between the barriers in the limit $b \to \infty$ and therefore generalized Hartman effect doesn't exist in SFQM. We observe that in SFQM, the tunneling time for large $b$ in the case of locally periodic potential is smaller than…
In this project, we will develop the foundations of quantum mechanics using the methods of supersymmetry. We will discuss the use of the superpotential to derive the supersymmetric partner of a potential in one dimension, and explore several key examples with an emphasis on shape invariant potentials. We will then discuss the modeling of supersymmetric quantum systems using matrices and operators, and how it relates to the eigenstate thermalization hypothesis.
Tunneling photons pose no challenge to Bohmian mechanics
本文指出，Sharoglazova等人实验中观测到的半经典速度与玻姆速度差异不构成对玻姆力学的挑战，因两者物理量本质不同，且两种解释在耦合波导中预测的光子隧穿动力学完全一致。
How much time does a wave packet spent in tunneling a barrier? Quantum mechanical calculations result in zero time inside a barrier. In the nineties analogous tunneling experiments with microwaves were carried out confirming quantum mechanics. Electron tunneling time is hard to measure being extremely short. However, quite recently the atomic ionization tunneling time has been measured. Experimental data of photonic, phononic, and electronic tunneling time is available now. It appears that the tunneling time is a universal property in the investigated time range of twelve orders of magnitude.
Shegelski, Kavka, and Hynbida have developed a method for calculating the lifetime of a particle initially localized in a potential well exactly quantum mechanically by employing a heuristic expression for the lifetime <t>. Their method allows for the inclusion of a bound state, and their results for tunneling through a centrifugal barrier demonstrate the major role that bound states play in determining the lifetime. In this comment, I show that one can greatly simplify the analysis by calculating <t^2>^(1/2) instead. Results obtained are exact when no bound state is present. In the presence o…
The study of the quantum to classical transition is of fundamental as well as technological importance, and focusses on mesoscopic devices, with a size for which either classical physics or quantum physics can be brought to dominate. A particularly diverse selection of such devices is available in cavity quantum-optomechanics. We show that these can be leveraged for the study of dynamical-tunnelling in a quantum chaotic system. This effect probes the quantum to classical transition deeply, since tunnelling rates sensitively depend on the ability of the quantum system to resolve the underlying …
This paper revisited quantum tunneling dynamics through a square double-barrier potential. We emphasized the similarity of tunneling dynamics through double-barrier and that of optical Fabry--P$\acute{e}$rot (FP) interferometer. Based on this similarity, we showed that the well-known resonant tunneling can also be interpreted as a result of matter multi-wave interference, analogous to that of FP interferometer. From this analogy, we also got an analytical finesse formula of double-barrier. Compared with that obtained numerically for a specific barrier configuration, we found that this formula …
Tunneling in double-well potentials within Nelson's stochastic mechanics: Application to ammonia inversion
本文利用Nelson随机力学研究双井势中束缚态的隧道时间统计，通过首次通过时间理论计算均值和概率分布，并验证了随机力学与量子力学隧道时间的关系，应用于氨分子翻转动力学。
Field emission tunnelling as a window onto fundamental issues in quantum mechanics
本文探讨了场发射隧穿在量子力学基本问题中的意义，包括电子本质、语言表述、数学用途及时间箭头等核心问题。
How much time does a tunneling wave packet spent in traversing a barrier? Quantum mechanical calculations result in zero time inside a barrier . In the nineties analogous tunneling experiments with microwaves were carried out. The results agreed with quantum mechanical calculations. Electron tunneling time is hard to measure being extremely short and parasitic effects due to the electric charge of electrons may be dominant. However, quite recently the atomic ionization tunneling time has been measured. Experimental data of photonic, phononic, and electronic tunneling time is available now and …
The concepts of quantile position, trajectory, and velocity are defined. For a tunneling quantum mechanical wave packet, it is proved that its quantile position always stays behind that of a free wave packet with the same initial parameters. In quantum mechanics the quantile trajectories are mathematically identical to Bohm's trajectories. A generalization to three dimensions is given.
In this paper we study the tunneling using a background independent (polymer) quantization scheme. We show that at low energies, for the tunneling through a single potential barrier, the polymer transmission coefficient and the polymer tunneling time converge to its quantum-mechanical counterparts in a clear fashion. As the energy approaches the maximum these polymer quantities abruptly decrease to zero. We use the transfer matrix method to study the tunneling through a series of identical potential barriers. We obtain that the transmission coefficients (polymer and quantum-mechanical) behave …
Hartman Effect from a Geometrodynamic Extension of Bohmian Mechanics
本文通过几何动力学扩展布雷姆-玻姆力学，揭示量子隧穿与时空几何的联系，解释哈特曼效应的几何自调节机制。
We study the quantum tunnel effect through a potential barrier employing a semiclassical formulation of quantum mechanics based on expectation values of configuration variables and quantum dispersions as dynamical variables. The evolution of the system is given in terms of a dynamical system for which we are able to determine effective trajectories for individual particles, in a total resemblance of the Bohmian description of quantum mechanics. We obtain a type of semiclassical confinement for particles in a similar way as with the quantum potential, and also determine a semiclassical transmis…
Quantum tunneling is considered from the point of view of local realism. It is concluded that a quantum object tunneling through a potential barrier cannot be interpreted as a point-like particle because such an interpretation generates a contradiction with the impossibility of faster-than-light motion. Such a contradiction does not arise if a quantum object is considered as a continuous medium formed by the fields of matter. The dynamics law of the mechanical motion of these matter fields is derived from the quantum evolution law in the path integrals form. The analysis of tunneling shows tha…
A relativistic quantum-mechanical description of guided waves is given, based on which we present an alternative way to describe and interpret the propagation of electromagnetic wave packets through an undersized waveguide. In particular, we show that the superluminal phenomenon of evanescent modes is actually a known conclusion in quantum field theory, and it preserves a quantum-mechanical causality.
Beginners studying quantum mechanics are often baffled with electron tunneling.Hence an easy approach for comprehension of the topic is presented here on the basis of uncertainty principle.An estimate of the tunneling time is also derived from the same method.
Scattering and Tunneling in Real Quantum Mechanics
该研究建立真实量子力学的演化方程，推导其演化算子性质，构建对应散射理论，证明其微分散射截面、隧穿概率与标准复量子力学结果一致。
In this work we present a theoretical model supported with a physical reasoning leading to a relation which performs an excellent estimation for the tunneling time in attosecond and strong field experiments, where we address the important case of the He-atom \cite{Eckle:2008s,Eckle:2008}. Our tunneling time estimation is found by utilizing the time-energy uncertainty relation and represents a quantum clock. The tunneling time is also featured as the time of passage (at the exit of the tunnel) similarly to the Einstein's {\it photon box Gedanken experiment}. Our work tackles an important study …
After the review by Hauge and Stovneng the old question of "How long does it take to tunnel through the barrier?" has not still lost its relevance. As before, there is no clear answer to this question even for the one-dimensional completed scattering (OCS). In this paper we show that this seemingly simple question stands alongside with such fundamental problems of quantum mechanics (QM) as the Schrodinger's-cat and and EPR-Bohm paradoxes. Their common feature is that the states of a scattered particle, a radioactive atom and electron EPR-pair represent pure {\it micro-cat} states. It is widely…
We explore the features of non-relativistic quantum tunneling in space fractional quantum mechanics through a family of Cantor potentials. We consider two types of potentials: general Cantor and general Smith-Volterra-Cantor potential. The Cantor potential is an example of fractal potential while the Smith-Volterra-Cantor potential doesn't belong to the category of a fractal system. The present study brings for the first time, the study of quantum tunneling through fractal potential in fractional quantum mechanics. We report several new features of scattering in the domain of space fractional …
By using an exact analytical non-Hermitian formalism involving the full set of resonance (quasinormal) states and complex energy eigenvalues for quantum tunneling decay, we show that unitarity holds at any instant of time for the nonstationary decaying probability density in the whole space, thus providing a fundamental description of the quantum mechanical decay process.
New prescription to treat position and time equally in quantum mechanics is presented. Using this prescription, we could successfully derive some interesting formulae such as time-of-arrival for a free particle and quantum tunneling formula. The physical interpretation will be discussed.
The conundrum parameter-operator of time in quantum mechanics (QM), as well as the time-energy uncertainty relation and the tunneling delay time, have recently been addressed in attosecond optical ioniza- tion experiments. The parameter status of time in the time dependent Schrödinger equation (TDSE) is supported by the well-known Pauli's ob- jection as well as by its interpretation as an emerging property of entangle- ment with a classical environment. On the other hand, the introduction of a self-adjoint dynamical time operator in Dirac's formulation of elec- tron's relativistic quantum mech…
We show that scattering a quantum particle on a one-dimensional potential barrier as well as scattering the electromagnetic wave on a quasi-one-dimensional layered structure (both represent scattering problems with one 'source' and two 'sinks') violate the superposition principle; the role of nonlinear elements is played here by the potential barrier and the layered structure, splitting the incident (probability and electromagnetic) wave into two parts (transmitted and reflected). This explains why all attempts to solve the tunneling time problem within the framework of the standard (linear) m…
Quantum mechanical objects or nanoobjects have been proposed as bits for information storage. While time-averaged properties of magnetic, quantum-mechanical particles have been extensively studied experimentally and theoretically, experimental investigations of the real time evolution of magnetization in the quantum regime were not possible until recent developments in pump-probe techniques. Here we investigate the quantum dynamics of effective spin systems by means of analytical and numerical treatments. Particular attention is paid to the quantum revival time and its relation to the magnetiz…
We reformulate quantum tunneling in a multi-dimensional system where the tunneling sector is non-linearly coupled to oscillators. The WKB wave function is explicitly constructed under the assumption that the system was in the ground state before tunneling. We find that the quantum state after tunneling can be expressed in the language of the conventional in-in formalism. Some implications of the result to cosmology are discussed.
Quantum mechanics predicts an exponentially small probability that a particle with energy greater than the height of a potential barrier will nevertheless reflect from the barrier in violation of classical expectations. This process can be regarded as tunneling in momentum space, leading to a simple derivation of the reflection probability.
Quantum mechanical real-time tunneling through general scattering potentials is studied in the semiclassical limit. It is shown that the exact path integral of the real-time propagator is dominated in the long time sector by quasi-stationary fluctuations associated with caustics. This leads to an extended semiclassical propagation scheme for wave packet dynamics which accurately describes deep tunneling through static and, for the first time, driven barrier potentials.
As an application of the polymer quantization scheme, in this work we investigate the one dimensional quantum mechanical tunneling phenomenon from the perspective of polymer representation of a non-relativistic point particle and derive the transmission and reflection coefficients. Since any tunneling phenomenon inevitably evokes a tunneling time we attempt an analytical calculation of tunneling times by defining an operator well suited in discrete spatial geometry. The results that we come up with hint at appearance of the Quantum Zeno Effect in polymer framework.
The tunneling effect is the most popular phenomenon of quantum physics and is present in modern physical theories. Still, the most important features of this effect are already present in toy models - low dimensional quantum mechanics with high potential barriers. Even these simple systems cannot be solved analytically and approximations have to be made. Tunneling is closely related to splitting of the ground state energy which is degenerate in the classical limit. In this thesis we use two methods to estimate the energy splitting in one dimensional quantum mechanics. In semi-classical approxi…
In recent years there has been a resurgence of interest in Bohmian mechanics as a numerical tool because of its local dynamics, which suggest the possibility of significant computational advantages for the simulation of large quantum systems. However, closer inspection of the Bohmian formulation reveals that the nonlocality of quantum mechanics has not disappeared -- it has simply been swept under the rug into the quantum force. In this paper we present a new formulation of Bohmian mechanics in which the quantum action, S, is taken to be complex. This leads to a single equation for complex S, …
We study the classical dynamics of non-relativistic particles endowed with spin. Non-vanishing Zitterbewegung terms appear in the equation of motion also in the small momentum limit. We derive a generalized work-energy theorem which suggests classical interpretations for tunnel effect and quantum potential.
Usually tunneling is established after imposing some matching conditions on the (time-independent) wave function and its first derivative at the boundaries of a barrier. Here an alternative scheme is proposed to determine tunneling and estimate transmission probabilities in time-dependent problems, which takes advantage of the trajectory picture provided by Bohmian mechanics. From this theory a general functional expression for the transmission probability in terms of the system initial state can be reached. This expression is used here to analyze tunneling properties and estimate transmission…
Quantum tunneling is a fundamental quantum mechanical effect involved in plenty of physical phenomena. Its control would impact these phenomena and the technologies based on them. We show that the quantum tunneling probability through a potential barrier can be increased to approach unity in a quantum Zeno dynamics undergone by the tunneling particle in which the direction of the momentum is frequently monitored. We first model the measurements of the momentum direction as selective von Neumann projections, and then as nonselective, direction-sensitive interactions of the particle with probe p…
We study the tunneling through delta and double delta potentials in fractional quantum mechanics. After solving the fractional Schrödinger equation for these potentials, we calculate the corresponding reflection and transmission coefficients. These coefficients have a very interesting behaviour. In particular, we can have zero energy tunneling when the order of the Riesz fractional derivative is different from 2. For both potentials, the zero energy limit of the transmission coefficient is given by $\mathcal{T}_0 = \cos^2{π/α}$, where $α$ is the order of the derivative ($1 < α\leq 2$).
Discrete breathers (DBs) -- a spatial time-periodic localization of energy -- are predicted in a large variety of non-linear systems. Motivated by the conceptual bridging of the DBs phenomena in classical and quantum mechanical representation, we study their signatures in the dynamics of a quantum equivalent of a classical mechanical point in a phase space -- a coherent state. We show that in contrast to a point in a phase space that can exhibit either delocalized or localized motion, a coherent state can show signatures of a quantum equivalent of both localized and delocalized behavior. In cl…
Static-Field Tunneling Ionization in Space-Fractional Quantum Mechanics
本文在空间分数量子力学中发展了类似ADK的隧穿模型，推导了三角势垒的闭合形式隧穿指数，改变了传统I_p^{3/2}的标度关系，并引入了sin(π/α)因子以编码非局域色散相关的复相位结构。
Tunneling time in non-Hermitian space fractional quantum mechanics
本研究首次探讨了非厄米空间分数阶量子力学中波包通过复势的隧穿时间，利用稳相法推导出闭式表达式，并揭示了吸收分量与Levy指数的相互作用可引发Hartman效应。
The quantum transition between shifted zero-mode wave functions is shown to be induced by the systematic deformation of topological and non-topological defects that support the $1$-dim double-well (DW) potential tunneling dynamics. The topological profile of the zero-mode ground state, $ψ_{0}$, and the first excited state, $ψ_{1}$, of DW potentials is obtained through the analytical technique of topological defect deformation. Deformed defects create two inequivalent topological scenarios connected by a symmetry breaking that support the quantum conversion of a zero-mode stable vacuum into an …
Quantization-based Optimization with Perspective of Quantum Mechanics
本文基于薛定谔方程分析基于量子化的优化，揭示隧穿效应能逃离局部最小值以实现全局优化，并在多模态基准函数上验证了该分析的有效性。
Quantum tunneling introduces a fundamental difference between classical and quantum mechanics. Whenever the classical ground state is non-unique (degenerate), quantum mechanics restore uniqueness thanks to tunneling. A condensate in a rotating trap with a vortex can have such a degenerate classical ground state, a degeneracy that is excluded in the absence of rotation at least when the Gross-Pitaevskii equation applies. If the rotating trap has a center of symmetry, like a figure eight (a peanut), the vortex may be on either side with the same energy yielding a degenerate ground state, a degen…
In a previous paper [Blado G, Owens C, and Meyers V 2014 Quantum Wells and the Generalized Uncertainty Principle Eur. J. Phys. 35 065011], we showed that quantum gravity effects can be discussed with only a background in non-relativistic quantum mechanics at the undergraduate level by looking at the effect of the generalized uncertainty principle (GUP) on the finite and infinite square wells. In this paper, we derive the GUP corrections to the tunneling probability of simple quantum mechanical systems which are accessible to undergraduates (alpha decay, simple models of quantum cosmogenesis an…
We adapt the semiclassical technique, as used in the context of instanton transitions in quantum field theory, to the description of tunneling transmissions at finite energies through potential barriers by complex quantum mechanical systems. Even for systems initially in their ground state, not generally describable in semiclassical terms, the transmission probability has a semiclassical (exponential) form. The calculation of the tunneling exponent uses analytic continuation of degrees of freedom into a complex phase space as well as analytic continuation of the classical equations of motion i…
Quantum Tunneling in SO(3,1)$\times$SO(4)
该研究用新隧穿变换T(v)补充洛伦兹变换L(v)，将其应用于初等量子力学以直观理解量子隧穿本质，并探讨了克莱因悖论。
Quantum mechanical tunneling across smooth double barrier potentials modeled using Gaussian functions, is analyzed numerically and by using the WKB approximation. The transmission probability, resonances as a function of incident particle energy, and their dependence on the barrier parameters are obtained for various cases. We also discuss the tunneling time, for which we obtain generalizations of the known results for rectangular barriers.
The anti self-adjoint operators of coordinate and momentum are introduced and applied in discussion of tunneling through the potential barrier where imaginary value of momentum unavoidably appears. Tunneling through temporal barrier is proceeded in a similar way and it is shown that the quantum system can tunnel, while the classical systems are destroyed by such barrier. The imaginary observables of coordinate and momentum are used in a novel treatment of passage through the event horizon. By considering the event horizon as the border of the finite potential barrier, it is found that there is…
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…
We consider in detail the quantum-mechanical problem associated with the motion of a one-dimensional particle under the action of the double-well potential. Our main tool will be the euclidean (imaginary time) version of the path-integral method. Once we perform the Wick rotation, the euclidean equation of motion is the same as the usual one for the point particle in real time, except that the potential at issue is turned upside down. In doing so, our double-well potential becomes a two-humped potential. As required by the semiclassical approximation we may study the quadratic fluctuations ove…
Inspired by new trends in atomtronics, cold atoms devices and Bose-Einstein condensate dynamics, we apply a general technique of N=4 extended Supersymmetric Quantum Mechanics to isospectral Hamiltonians with triple-well potentials, i.e. symmetric and asymmetric. Expressions of quantum-mechanical propagators, which take into account all states of the spectrum, are obtained, within the N = 4 SQM approach, in the closed form. For the initial Hamiltonian of a harmonic oscillator, we obtain the explicit expressions of potentials, wavefunctions and propagators. The obtained results are applied to tu…
We propose an electromechanical transducer based on a resonant-tunneling configuration that, with respect to the standard tunneling transducers, allows larger tunneling currents while using the same bias voltage. The increased current leads to an increase of the shot noise and an increase of the momentum noise which determine the quantum limit in the system under monitoring. Experiments with micromachined masses at 4.2 K could show dominance of the momentum noise over the Brownian noise, allowing observation of the quantum-mechanical noise at the mesoscopic scale.
Chaos and Quantum Tunneling
本文探讨了非完全可积也不完全混沌的哈密顿系统中混沌与量子隧穿的关系，分析了混沌辅助和共振辅助隧穿现象，并讨论了混沌对隧穿概率的增强机制。
Despite quantum tunneling has been studied since the advent of quantum mechanics, the literature appears to contain no simple (textbook) formula for tunneling in generic asymmetric double-well potentials. In the regime of strong localization, we derive an succinct analytical formula based on the WKB semi-classical approach. Two different examples of asymmetric potentials are discussed: when the two localized levels are degenerate or not. For the first case, we also discuss a time-dependent problem showing quantum Zeno effect.
The non-perturbative renormalization group (NPRG) is applied to analysis of tunnelling in quantum mechanics. The vacuum energy and the energy gap of anharmonic oscillators are evaluated by solving the local potential approximated Wegner-Houghton equation (LPA W-H eqn.). We find that our results are very good in a strong coupling region, but not in a very weak coupling region, where the dilute gas instanton calculation works very well. So it seems that in analysis of quantum tunnelling, the dilute gas instanton and LPA W-H eqn. play complementary roles to each other. We also analyze the supersy…
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