We consider the time development of a one-dimensional Gaussian wave packet. Noting that the quantum mechanical predictions concerning position and momentum measurements of a free particle described by a Gaussian packet are consistent with the predictions one would make on the basis of a classical “Naive Model,” we investigate whether this model can account for the scattering of a particle by a rectangular barrier. Obtaining numerical solutions of the time dependent Schrödinger equation for a series of packets with varying incident momenta, we find that the Naive Model is inadequate to explain the quantum mechanical transmission. We study the relative speeds of the transmitted and reflected packets, discussing a number of possible interpretations and indicating the quantum mechanical result. Finally, we note that for certain choices of the parameters describing the packet and the barrier, the reflected packet is composed of two distinct humps.
We introduce a formalism for the calculation of the time of arrival t at a detector of particles traveling through interacting environments. We develop a general formulation that employs quantum canonical transformations from the free to the interacting cases to compute t. We interpret our results in terms of a Positive Operator Valued Measure. We then compute the probability distribution in times of arrival at a detector of those particles that, after their initial preparation, have undergone quantum tunneling or reflection due to the presence of potential barriers. We obtain the expected retardation or advancement for transmitted wave packets, and non-foreseen double bump structures for some cases of reflection.
The classical limit problem of quantum mechanics is revisited on the basis of a scheme that enables a quantitative study of the way the quantum-classical agreement emerges while going through the intermediate mass range between the microscopic and the macroscopic domains. As a specific application of such a scheme, we investigate the classical limit of a quantum time distribution - an area of study that has remained largely unexplored. For this purpose, we focus on the arrival time distribution in order to examine the way the observable results pertaining to the quantum arrival time distribution which is defined in terms of the probability current density gradually approach the relevant classical statistical results for an ensemble that corresponds to a Gaussian wave packet evolving in a linear potential.
We model ideal arrival-time measurements for free quantum particles and for particles subject to an external interaction by means of a narrow and weak absorbing potential. This approach is related to the operational approach of measuring the first photon emitted from a two-level atom illuminated by a laser. By operator normalizing the resulting time-of-arrival distribution, a distribution is obtained which for freely moving particles not only recovers the axiomatically derived distribution of Kijowski for states with purely positive momenta but is also applicable to general momentum components. For particles interacting with a square barrier the mean arrival time and corresponding ``tunneling time'' obtained at the transmission side of the barrier become independent of the barrier width (Hartman effect) for arbitrarily wide barriers, i.e., without the transition to the ultraopaque, classical-like regime dominated by wave packet components above the barrier.
In a recent paper Grot, Rovelli, and Tate (GRT) [Phys. Rev. A 54, 4676 (1996)] derived an expression for the probability distribution $\ensuremath{\pi}(T;X)$ of intrinsic arrival times
T(X) at position
x=X for a quantum particle with initial wave function $\ensuremath{\psi}(x,t=0)$ freely evolving in one dimension. This was done by quantizing the classical expression for the time of arrival of a free particle at
X, assuming a particular choice of operator ordering, and then regulating the resulting time of arrival operator. For the special case of a minimum-uncertainty-product wave packet at
t=0 with average wave number
〈k〉 and variance $\ensuremath{\Delta}k$ they showed that their analytical expression for $\ensuremath{\pi}(T;X)$ agreed with the probability current density
J(x=X,t=T) only to terms of order $\ensuremath{\Delta}k/〈k〉.$ They dismissed the probability current density as a viable candidate for the exact arrival time distribution on the grounds that it can sometimes be negative. This fact is not a problem within Bohmian mechanics where the arrival time distribution for a particle, either free or in the presence of a potential, is rigorously given by
∣J(X,T)∣ (suitably normalized) [W. R. McKinnon and C. R. Leavens, Phys. Rev. A 51, 2748 (1995); C. R. Leavens, Phys. Lett. A 178, 27 (1993); M. Daumer et al., in On Three Levels: The Mathematical Physics of Micro-, Meso-, and Macro-Approaches to Physics, edited by M. Fannes et al. (Plenum, New York, 1994); M. Daumer, in Bohmian Mechanics and Quantum Theory: An Appraisal, edited by J. T. Cushing et al. (Kluwer Academic, Dordrecht, 1996)]. The two theories are compared in this paper and a case presented for which the results could not differ more: According to GRT's theory, every particle in the ensemble reaches a point
x=X, where $\ensuremath{\psi}(x,t)$ and
J(x,t) are both zero for all
t, while no particle ever reaches
X according to the theory based on Bohmian mechanics. Some possible implications are discussed.