A Time Dependent Version of the Quantum WKB Approximation

In this paper, the tunnelling of a particle through a potential barrier is investigated in the presence of a time-dependent perturbation. The latter is attributed to the process of the energy measurement of the scattered particle. The time-dependent Schrodinger equation of the model is exactly solved. The calculation of the probability density inside the barrier proves that the tunnelling dynamics is determined not only by the transmitted and reflected waves but also by their...

This article is a slightly expanded version of the talk I delivered at the Special Plenary Session of the 46-th Annual Meeting of the Israel Physical Society (Technion, Haifa, May 11, 2000) dedicated to Misha Marinov. In the first part I briefly discuss quantum tunneling, a topic which Misha cherished and to which he was repeatedly returning through his career. My task was to show that Misha's work had been deeply woven in the fabric of today's theory. The second part is an a...

We present a new semiclassical method that yields an approximation to the quantum mechanical wavefunction at a fixed, predetermined position. In the approach, a hierarchy of ODEs are solved along a trajectory with zero velocity. The new approximation is local, both literally and from a quantum mechanical point of view, in the sense that neighboring trajectories do not communicate with each other. The approach is readily extended to imaginary time propagation and is particular...

In the B\"uttiker-Landauer perturbation approach to electron tunnelling, through a time-modulated rectilinear potential barrier, the Tien-Gordon identity was invoked, together with its infinite energy spectrum. Here, an exact treatment is presented which is based on the temporal wave-function matching procedure, that led to a finite energy spectrum. In seeking the condition governing the time evolution of the tunnelling process, the Euler formula provided the crucial ingredie...

Time evolution of tunneling phenomena in medium is studied using a standard model of environment interaction. A semiclassical formula valid at low, but finite temperatures is derived in the form of integral transform for the reduced Wigner function, and the tunneling probability in thermal medium is calculated for a general tunneling potential of one dimensional system. Effect of dissipation, its time evolution in particular, depends on the behavior of the potential far beyon...

A simple model of a quantum clock is applied to the old and controversial problem of how long a particle takes to tunnel through a quantum barrier. The model I employ has the advantage of yielding sensible results for energy eigenstates, and does not require the use of time-dependant wave packets. Although the treatment does not forbid superluminal tunneling velocities, there is no implication of faster-than-light signaling because only the transit duration is measurable, not...

A new formalism will be presented in order to study real time evolution of quantum systems at finite temperature. Probability distributions for time-correlated observables will be studied non-perturbatively and fully quantized. This works for various systems, including ones with tunneling phenomena. We have obtained good results with some computational methods which can be used on models with several degrees of freedom. Thus it looks feasible to study vacuum tunneling in real...

Addressed, mainly: postgraduates and related readers. Subject: Given two classical mechanical 1D-moving particles (material points), with identical initial data, one of those particles given free and another given to pass through a symmetrical force-barrier, a retardation effect is observed: After the barrier has been passed over, the second particle moves with the same velocity as the free particle, but spacially is retarded with respect to the latter. If the "non-free" part...

We derive an expression for the conditional time for the reflection of a wave from an arbitrary potential barrier using the WKB wavefunction in the barrier region. Our result indicates that the conditional times for transmission and reflection are equal for a symmetric barrier within the validity of the WKB approach.

We consider fundamental problems on the understanding of the tunneling phenomena in the context of the multi-dimensional wave function. In this paper, we reconsider the quantum state after tunneling and extend our previous formalism to the case when the quantum state before tunneling is in a squeezed state. Through considering this problem, we reveal that the quantum decoherence plays a crucial role to allow us of the concise description of the quantum state after tunneling.