We discuss the propagation of wave packets through interacting environments. Such environments generally modify the dispersion relation or shape of the wave function. To study such effects in detail, we define the distribution function
PX(T), which describes the arrival time T of a packet at a detector located at point X. We calculate
PX(T) for wave packets traveling through a tunneling barrier and find that our results actually explain recent experiments. We compare our results with Nelson's stochastic interpretation of quantum mechanics and resolve a paradox previously apparent in Nelson's viewpoint about the tunneling time.
The problem of tunneling time associated with the passage of a particle through a tunneling barrier seems very simple. But it turns out to be very deceptive by the lack of a time operator and there is no clear consensus about simple expression for the tunneling time. In this paper we evaluate the tunneling time based on Nelson's quantum stochastic process approach. This approach describes the motion of a particle by a stochastic differentia1 equation and gives a sample paths. The ensemble of the paths reproduce the predictions given by ordinary quantum mechanics in an average. Each sample path has its own time dependent history, which enables us make a virtual quantum experiment. We formulate the tunneling time based on this approach and compare it with the Larmor time estimated from an experimental measurement of Larmor precession angles of tunneling neutrons.
Nelson's stochastic mechanics formulates quantum dynamics as a real-time conservative diffusion process in which a particle undergoes Brownian-like motion with a fluctuation amplitude fixed by Planck's constant. While being mathematically equivalent to the Schrödinger formulation, this approach provides an alternative dynamical framework that enables the study of time-resolved quantities that are not straightforwardly defined within the standard operator-based approach. In the present work, Nelson's stochastic mechanics is employed to investigate tunneling-time statistics for bound states in double-well potentials. Using first-passage time theory within this framework, both the mean tunneling time,
τˉ, and the full probability distribution,
p(τ), are computed. The theoretical predictions are validated through extensive numerical simulations of stochastic trajectories for two representative potentials. For the square double-well potential, analytical expressions for
τˉ are derived and are shown to be in excellent agreement with simulations. In the high-barrier limit, the results reveal a direct relation between the stochastic-mechanical and quantum-mechanical tunneling times, expressed as
τQM=(π/2)τˉ, where
τQM corresponds to half the oscillation period of the probability of finding the particle in either well. This relation is further confirmed for generic double-well systems through a WKB analysis. As a concrete application, the inversion dynamics of the ammonia molecule is analyzed, yielding an inversion frequency of approximately 24 GHz, in close agreement with experimental observations. These results highlight the potential of stochastic mechanics as a conceptually coherent and quantitatively consistent framework for analyzing tunneling phenomena in quantum systems.
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.