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.
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.
The tunneling time of a particle through a given barrier is commonly defined in terms of “internal clocks” that effectively measure the interaction time with internal degrees of freedom of the barrier. It is known that this definition of the time scale for tunneling is not unique in the sense that it depends on the clock used to define it. For the case of resonance tunneling, a particular choice that in the limit of a high/broad square barrier yields the original result of Büttiker and Landauer ( Phys. Rev. Lett. 1982, 49, 1739) is correlated to the lifetime of the resonance state. This is illustrated for analytically solvable one-dimensional double barrier models and for a realistic model of electron tunneling through a static water barrier. The latter calculation constitutes a novel application of this concept to a 3-dimensional model, and the observed structure in the energy dependence of the computed traversal time reflects the existence of transient tunneling resonances associated with instantaneous water structures. These models, characterized by the existence of shape resonances in the barrier, make it possible to examine different internal clocks that were proposed for measuring tunneling times in situations where a “clock independent” intrinsic time scale (the resonance lifetime) for the tunneling time exists. It is argued that this time may be used in order to estimate the relative importance of dynamical barrier processes that affect the tunneling probability.
The wave-packet simulation (WPS) method for calculating the time a tunneling particle spends inside a one-dimensional potential barrier is reexamined using the Feynman path-integral technique. Following earlier work by Sokolovski and Baskin [Phys. Rev. A 36, 4604 (1987)], the tunneling (or traversal) time tpackT is defined as a matrix element of a classical nonlocal functional between two states that represent the initial and transmitted wave packets. These states do not lie on the same orbit in Hilbert space; as a result, tpackT is complex-valued. It is shown that RetpackT reduces to the standard WPS result, tphaseT, for conditions similar to those employed in the conventional WPS analysis. Similarly, ImtpackT is shown to contain information about the energy dependence of the transmission probability. Under semiclassical conditions, ImtpackT reduces to the well-known Wentzel-Kramers-Brillouin expression for the tunneling time. It is shown there are different definitions for the traversal time of a classical moving object, whose size is comparable to the width of the region of interest. In the quantum case, these different definitions correspond to different ways of analyzing the WPS experiment. The path-integral approach demonstrates that the tunneling-time problem is one of understanding the physical significance of complex-valued off-orbit matrix elements of an operator or functional. The physical content of complex-valued tunneling times is discussed. It is emphasized that the use of complex tunneling times includes real-time approaches as a special case. Nevertheless, there is a limitation in the description of tunneling experiments using tunneling times, whether real or complex. The path-integral approach does not supply a universal traversal time, analogous to a classical time, that can be used in quantum situations. It is demonstrated that the often expressed hope of finding a well-defined and universal real tunneling time is erroneous.
This Thesis consists of two related parts. The first part (Chapters 2 and 3) presents the results of a fundamental theoretical study of the interpretation of quantum mechanics, and particularly of tunneling processes. Part two (Chapters 4, 5, and 6) describes the experimental results of spectroscopic investigations of quantum mechanical interactions between chromophores, which may play a role in electron transport by means of tunneling.The formulation of the quantum theory at the beginning of the twentieth century resulted in a scientific revolution. The impact of this theory on our physical world picture is still visible today. The quantum theory is impressively successful in its predictions of the outcomes of experiments, but the interpretation of the theory is still shaky. The predictions of quantum mechanics about the outcome of experiments are detailed and testable, but the meaning of the theory for quantum systems, in which no measurement has taken place is not clear. Various interpretations exist. Two of these are compared in Chapter 2. The first is the orthodox or Copenhagen interpretation, which was advocated by the Copenhagen-based Niels Bohr. His followers claimed that, in the orthodox interpretation, nothing can be said about the properties of systems if no suitable measurement has been carried out. The alternative is the causal interpretation of quantum mechanics, in which a particle, e.g. an electron, has a well-defined position and velocity at each instant of time independent of a measurement act. The causal theory does not assign a special role to the observer. In the causal interpretation the location as a function of time amounts to trajectories. The two interpretations are empirically equivalent, i.e. they predict the same experimental outcomes.In Chapter 2 the meaning of the two interpretations is clarified by the notion of tunneling time. Tunneling is the quantum mechanical phenomenon that a particle can cross a barrier even if its energy is less than the barrier energy. The time a particle takes to tunnel is the tunneling time. Time is a confusing concept: quantum theory appears not to know how to handle this topic. Time appears in the quantum mechanical formulae as a parameter, and not as an operator that corresponds to an observable quantity. Since an unambiguous time operator is missing, it is interesting to see what different interpretations have to say about tunneling times. It is strictly speaking impossible to define tunneling times in the orthodox interpretation because of the lack of a clear time operator, of which the expectation value can be observed by experiment. In the causal interpretation, every trajectory corresponds to a possible path of an electron. The density functions determine the probability that particles move on particular trajectories. This is how the stochastic character of quantum mechanics is accounted for in this interpretation. The transmission time is defined as the time spent inside the barrier by the electrons that eventually cross the barrier. The average transmission time can be defined by means of the trajectories, both in the case of stationary wave functions and in that of time-dependent, Gaussian wave packets. The Gaussian wave packets travel from minus infinity to plus infinity. On the way they meet potential barriers, which cause them to be reflected or transmitted. The process by which an electron moves from one chromophore to another can be described as tunneling because of the high potential barrier formed by the medium between the chromophores. In this Thesis the experimental media used are polymer films and apolar solutions. The model described in Chapter 2 is generally used in the literature but appears not to be applicable for a system of two chromophores and a tunneling electron, because electrons belonging to chromophores neither start in infinity nor go to infinity. A better model, a double potential well, is used in Chapter 3. The conclusion of Chapter 2 is that the transmission time can be defined in the causal interpretation when time-dependent wave packets are used.Chapter 3 illustrates the preceding by tunneling of a particle from one potential well to another passing a potential barrier. This model is the so-called double potential well. Examples are primary charge separation in the photosynthetic reaction center and in artificial model systems of chromophores in solutions and polymer films. The latter two cases are investigated spectroscopically in this Thesis. In these molecular systems, effective electron transport between chromophores after absorption of light takes place, yielding charge separation, i.e. the formation of chromophore cations. During their transport between natural or artificial chromophores, the released electrons often have to pass weak conducting media, corresponding to high potential barriers. If the energy of the electron exceeds the energy of the potential barrier, electron transport can be explained by classical theories. However, if the energy of the electron falls below the energy of the potential barrier, this is no longer possible. The transport mechanism is then understood as quantum mechanical tunneling. Experimental proof of tunneling has been found at low temperatures in photosynthesis.In Chapter 3 the chromophores and the tunneling electrons are described by double potential wells. Each well represents a chromphore and the space between the molecules a potential barrier. The formalism of the causal interpretation is applied to define a transmission time. The transmission time can be found using trajectories. An alternative method provides the average transmission time without calculating the trajectories. Arrival time distribution functions are defined and these are used to determine the average arrival time of the tunneling electrons at the entrance and at the exit of the barrier. Because arrival time distribution functions and hence both average arrival times and transmission times are not determined in an experimental context, we conclude that these definitions of transmission times are meaningless in the orthodox interpretation. The question posed at the end of Chapter 3 is whether, and if so how, these transmission times are experimentally accessible.The theoretical models developed in Chapters 2 and 3 are too simple to be extrapolated to real systems, such as electron transport between an electron donor and an acceptor, one of which is excited by light. Intermolecular interactions between excited electron donor molecules and ground state surrounding molecules cannot be neglected in chromophore films and solutions with relatively high concentrations of chromophores. These excitonic interactions should be included in adequate descriptions of electron transfer. Chapters 4, 5, and 6 describe these interactions using absorption and emission spectroscopy. In Chapter 4 a theoretical description of the absorption characteristics of chromophores in disordered films is given. At low concentrations, chromophores in films are mainly monomers. On statistical grounds, however, we conclude that some molecules are close enough together to form dimer-like structures, concentration pairs with characteristic absorption spectra. The presence of these concentration pairs influences the emission characteristics of chromophore films as well. When monomer electrons are excited by the absorption of photons, the electrons might return to the ground state, giving off their energy as fluorescence. Light energy can also be transferred to neighboring concentration pairs, in which case the fluorescence is quenched.The aim of Chapter 4 is to estimate the influence of potential quenchers in the vicinity of excited monomers and to predict the influence of these quenchers on the absorption spectra. The transfer of energy from monomers to concentration pairs, the so-called Förster transfer, takes place over quite large distances (1-5 nm). Statistics can be used to calculate the number of quenchers in particular concentrations of chromophores. In the literature it is assumed that two monomers that form a concentration pair are randomly oriented with respect to each other. If these pairs consisted of randomly oriented monomers and all fluctuations are taken into account, it appears on theoretical grounds that there would not be enough quenchers to describe the experimental changes. Because most chromophores are elongated or pancake-like, they are not randomly oriented but exhibit preferential orientations. These preferential orientations increase the efficiency of fluorescence quenching and theory and experiment are thereby reconciled. An analytical expression of the spectral changes compared to monomeric spectra of randomly oriented dimers is presented. To describe the interaction between chromophores that have one dominant transition dipole moment, a point dipole exciton model is used. To enhance the description of the spectral changes, homogeneous and inhomogeneous line broadening and monomer-to- monomer distance fluctuations should be taken into account. Inclusion of these factors in the model gives rise to the conclusion that the dimer spectra are broadened compared to the monomeric spectra and that the broadening is stronger at the blue end than at the red end of the spectra. However, to detect the blue-shifted spectral profile of randomly oriented dime-like samples, the concentration needs to be so high that isolated dimers cannot be observed. Then, simulation of the dimers becomes impossible, and all mutual interactions must be taken into account. Modeling of these interactions shows that the blue shift is not visible, and that only a broadening of the absorption spectra with respect to the monomeric spectra remains.Chapter 5 describes spectral changes that result from changes in the concentration changes of elongated Erythrosin-B (Ery B) chromophores in polyvinylalcohol (PVA) films. At low concentrations, monomers ar
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.
The tunneling time of particle through given barrier is commonly defined in terms of "internal clocks" which effectively measure the interaction time with internal degrees of freedom of the barrier. It is known that this definition of the time scale for tunneling is not unique in the sense that it depends on the clock used to define it. For the case of resonance tunneling, a particular choice that in the limit of a high/broad square barrier yields the original result of Buttiker and Landauer (Phys. Rev. Lett. 1982, 49, 1739) is correlated to the lifetime of the resonance state. This is illustrated for analytically solvable one-dimensional double barrier models and for a realistic model of electron tunneling through a static water barrier. The latter calculation constitutes a novel application of this concept to a 3-dimensional model, and the observed structure in the energy dependence of the computed traversal time reflects the existence of transient tunneling resonances associated with instantaneous water structures. These models, characterized by the existence of shape resonances in the barrier, make it possible to examine different internal clocks that were proposed for measuring tunneling times in situations where a "clock independent" intrinsic time scale (the resonance life time) for the tunneling time exists. It is argued that this time may be used in order to estimate the relative importance of dynamical barrier processes that affect the tunneling probability.