The precise definition and physical interpretation of a tunneling time is a fundamental problem in quantum mechanics. The lack of a well-defined time operator in quantum theory precludes the calculation of time in terms of an expectation value. Consequently, model calculations and simulation studies have been proposed and used to determine the time for a particle to traverse a quantum-mechanical barrier. The results offer both qualitatively and quantitatively, disparate predictions. One of the more commonly used approaches is the phase method, which we have applied to several one-dimensional model potentials to show that the tunneling time is characterized by the ratio of the typical decay length in the barrier to the incident velocity, τ∼(1/κ)/v0. We also show that the phase and the spin precession methods are equivalent when the magnetic field is applied throughout the space containing the particle wave function and the tunneling barrier. In principle, the spin precession method can be regarded as an operational definition of the tunneling time. It also has been shown that a tunneling time can be defined operationally for a time-dependent barrier, for which the model calculations predict a tunneling time given by the mean barrier width divided by the magnitude of the imaginary velocity of the particle in the barrier, τ∼d/‖v‖. This pseudokinematic result has been inferred by magnetic field dependent tunneling through heterostructures and computer simulation of wave packet tunneling through one-dimensional square barriers. Other tunneling time theories predict one or the other of the above mentioned characteristic times; several theories predict both. We interpret these different results for τ to mean that a tunneling time is not uniquely defined as a function of particle energy and the barrier shape unless the process in which the tunneling is involved is also specified. That is, the tunneling time is a meaningful and unique concept only for particular tunneling processes or definite operational procedures.
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.
A Brief History of Quantum Tunneling Some Basic Questions Concerning Quantum Tunneling Simple Solvable Problems Time-Dependence of the Wave Function in One-Dimensional Tunneling Semiclassical Approximations Generalization of the Bohr - Sommerfeld Quantization Rule and its Application to Quantum Tunneling Gamow's Theory, Complex Eigenvalues, and the Wave Function of a Decaying State Tunneling in Symmetric and Asymmetric Local Potentials and Tunneling in Nonlocal and Quasi-Solvable Barriers Classical Descriptions of Tunneling Tunneling in Time-Dependent Barriers Decay Width and Scattering Theory The Method of Variable Reflection Amplitude Applied to Solve Multichannel Tunneling Problems Path Integral and Its Semi-Classical Approximation in Quantum Tunneling Heisenberg's Equations of Motion for Tunneling Wigner Distribution Function in Quantum Tunneling Decay Widths of Siegert States, Complex Scaling and Dilatation Transformation Multidimensional Quantum Tunneling Group and Signal Velocities Time-Delay, Reflection Time Operator and Minimum Tunneling Time More about Tunneling Time Tunneling of a System with Internal Degrees of Freedom Motion of a Particle in a Waveguide with Variable Cross Section and in a Space Bounded by a Dumbbell-Shaped Object Relativistic Formulation of Quantum Tunneling Inverse Problems of Quantum Tunneling Some Examples of Quantum Tunneling in Atomic and Molecular Physics Some Examples from Condensed Matter Physics Alpha Decay.
Various approaches have been used to calculate the quantum-mechanical tunneling time through potential barriers including the phase-delay method first introduced by Bohm and Wigner, Buttiker's analysis of the Larmor clock and its generalization, the dwell time of Smith, and numerical studies of wavepacket propagation through potential barriers among others. Most of those previous estimates have only dealt with the tunneling time through simple obstacles, including delta-potential scatterers and simple rectangular barriers under zero bias condition. Even for these simple cases, the agreement among the various estimates is far from being satisfactory. In this manuscript, the transmission line technique is employed to solve the time-dependent Schrodinger equation and an expression of the quantum-mechanical tunneling time is derived for an arbitrary potential profile under non zero bias condition. An exact analytical expression of the tunneling time through a rectangular barrier is derived and shown to be identical to the one obtained recently by Spiller et. al. using the Bohm's quantum potential approach. The tunneling time through resonant tunneling structures under zero bias is also calculated and is shown to be minimum at the quasi-boundstate energy. Finally, the quantum-mechanical tunneling time through the emitter-base junction of a typical heterojunction bipolar transistor is shown to be larger than its semiclassical counterpart.
Analytic expressions are given for the time spent by a particle tunneling through a potential barrier. The expressions are derived for an incident wave packet which is initially Gaussian, centered about a point an arbitrary distance away from a rectangular potential barrier and moving toward the barrier with constant average velocity. Upon collision with the barrier, the packet splits into a transmitted and a reflected packet. The resultant transmission time is positive, nonzero and in principle measurable. Although the transmission time becomes quite large as the incident kinetic energy becomes very small, in general, for nonzero incident momentum and finite potential barriers which are neither very thick nor very thin, the transmission times are less than the time that would be required for the incident particle to travel a distance equal to the barrier thickness. The transmission times for metal-insulator-metal thin film sandwiches, given approximately by δt3=ℏ/(Efφ)12, where Ef is the Fermi energy of the metal, and φ the vacuum work function, are of the order of 10−16 sec, compared to RC time constants of about 10−13 sec.
ABSTRACT In the last decade, the rapid evolution of ultrafast laser spectroscopy has provided new impetus to the branch of chemistry dealing with tunneling processes involving electrons and light atoms. The possibility of measuring the time required by a particle to cross a barrier opens new perspectives for understanding the kinetics and dynamics of tunneling processes. In this study, a model is developed to estimate the tunneling time in tautomerization reactions based on proton transfer through a classically forbidden barrier. The proposed approach uses the asymptotic values of the wavefunction to compute the dwell time, which is the average time spent by the proton inside the potential barrier. The theory is formulated for both static barriers, which are good approximations for isolated molecules, and dynamic barriers to account for the effects of the chemical bath. The model performance is evaluated by two numerical simulations, in which a proton is transferred via tunneling in a double‐well potential; the tunneling barrier is approximated by a static and dynamical rectangular potential. A possible extension of the theory to open systems is briefly discussed within the framework of the Caldeira–Leggett model.