Delay time is usually discussed in terms of a wavepacket going through a potential barrier and its time of reappearance on the other side of it, as compared with the corresponding problem in free motion. Wigner stressed though that an interaction should be described by an matrix and a particular case would be that of a single resonance. Long ago a dynamical model was developed for this case in which the interaction takes place at a point through boundary conditions. It is used here to discuss the passage to x > 0 of a plane wave initially constrained to the interval . Comparing our result with the corresponding problem of free-particle motion, at the classical time of flight of the particles, we obtain information on delay times for wide and narrow resonances. Note that our approach is different from the one using stationary states, i.e. when the time , originally introduced by Wigner. For completeness we also discuss the latter for our problem in the concluding section and compare the results with those obtained in this paper.
How much time does it take for a particle to tunnel through a barrier? This question continues to baffle up to this very day, one of the reasons being the non-existence of an energy–time commutator analogous to the position–momentum commutation relation. Instead of worrying about the definition of an operator one may consider time as a parameter in the time-dependent Schrödinger equation and study the time evolution of a wavepacket appropriately scattered on a potential barrier. Using this approach, one may formally define a tunnelling flight time that may be measured in a time-of-flight experiment. The resulting tunnelling flight time either vanishes or is very small. The implications of this observation with respect to recent hydrogen and helium atom attosecond photoionization experiments are discussed. The vanishing or small flight time does not contradict the finite time measured in Larmor clock experiments, where the tunnelling particle affects an external degree of freedom whose dynamics induced by the tunnelling may be interpreted in terms of a time scale. Flight times are also relevant to quantum reflection where coherences due to resonance scattering are not well accounted for by Wigner's phase time delay.
We consider an analytic solution of the time-dependent Schr\"odinger equation with the initial condition \ensuremath{\psi}(x,0)=exp(ikx) along -\ensuremath{\infty}0 to investigate the time evolution for x>0 of the wave function in a double-barrier resonant structure at resonance. For typical parameters of the structure we find that the single-resonance approximation is valid from a few tenths of the corresponding lifetime onward. Very short times require the contribution of many far away resonances. The buildup time along the internal region takes a few lifetimes. At birth the transmitted wave front is blurred; however, for long times it becomes well defined and moves with classical velocity yielding a delay time \ensuremath{\tau}\ensuremath{\approx}2\ensuremath{\hbar}/\ensuremath{\Gamma} as in the stationary-phase treatment.
We present stationary expressions for the on-shell time delay operator and for its trace in quantum-mechanical potential scattering. The trace of this operator is shown to represent the time delay for a monoenergetic and well-collimated beam of initial states. For exponentially decaying potentials the total scattering cross section and the trace of the on-shell time delay operator have analytic extensions to complex energies, with poles at most at the usual resonance poles (poles of the analytically continued S -matrix) in the lower half plane and at their mirror images with respect to the real axis. At real energies close to such a pole both the scattering cross section and the time delay are approximately of Breit–Wigner type, the sign of the time delay being related to that of the virial. These results corroborate the idea that a resonance should show up through a peak in the scattering cross section and a large (though positive) time delay.
The time it takes a particle to tunnel through the asymmetric Eckart barrier potential is investigated using Gaussian wavepackets, where the barrier serves as a model for the potential along a chemical reaction coordinate. We have previously shown that the, in principle experimentally measurable, tunneling flight time, which determines the time taken by the transmitted particle to traverse the barrier, vanishes for symmetric potentials like the Eckart and square barrier [ Petersen , J. ; Pollak , E. J. Phys. Chem. Lett. 2017 , 9 , 4017 ]. Here we show that the same result is obtained for the asymmetric Eckart barrier potential, and therefore, the zero tunneling flight time seems to be a general result for one-dimensional time-independent potentials. The wavepacket dynamics is simulated using both an exact quantum mechanical method and a classical Wigner prescription. The excellent agreement between the two methods shows that quantum coherences are not important in pure one-dimensional tunneling and reinforces the conclusion that the tunneling flight time vanishes.
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 Re
tpackT reduces to the standard WPS result,
tphaseT, for conditions similar to those employed in the conventional WPS analysis. Similarly, Im
tpackT is shown to contain information about the energy dependence of the transmission probability. Under semiclassical conditions, Im
tpackT 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.