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Physical Review Letters

Beyond the Schrödinger Equation: Quantum Motion with Traversal Time Control

D. Sokolovski

Published Dec 22, 1997• 27 citations
Publisher

Abstract

We study a quantum particle, for which the duration $\ensuremath{\tau}$ it spends in some region of space is controlled by a meter, e.g., a Larmor clock. The particle is described by a wave function $\ensuremath{\Psi}(x,t\ensuremath{\mid}\ensuremath{\tau})$, with $|\ensuremath{\Psi}(x,t\ensuremath{\mid}\ensuremath{\tau}){|}^{2}$ giving the distribution of the meter's readings at location $x$. The wave function satisfies the ``clocked'' Schr\"odinger equation, which we solve numerically for the cases of bound motion and wave packet scattering. The method is shown to be a natural extension of the conventional quantum mechanics.

DOI
10.1103/physrevlett.79.4946

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We study a quantum particle, for which the duration \ensuremathτ\ensuremath{\tau}\ensuremathτ it spends in some region of space is controlled by a meter, e.g., a Larmor clock. The particle is described by a wave function \ensuremathΨ(x,t\ensuremath∣\ensuremathτ)\ensuremath{\Psi}(x,t\ensuremath{\mid}\ensuremath{\tau})\ensuremathΨ(x,t\ensuremath∣\ensuremathτ), with ∣\ensuremathΨ(x,t\ensuremath∣\ensuremathτ)∣2|\ensuremath{\Psi}(x,t\ensuremath{\mid}\ensuremath{\tau}){|}^{2}∣\ensuremathΨ(x,t\ensuremath∣\ensuremathτ)∣2 giving the distribution of the meter's readings at location xxx. The wave function satisfies the ``clocked'' Schr"odinger equation, which we solve numerically for the cases of bound motion and wave packet scattering. The method is shown to be a natural extension of the conventional quantum mechanics.
We solve a classical analog of the quantum traversal time problem. The classical diffusion equation is modified to control the amount of time spent by a particle in a specified region of space. The ``clocked'' diffusion equation is solved for an ensemble of Brownian particles confined in a closed volume. Long term behavior of the exposure time distribution is analyzed. A general recipe for including time control into fieldlike equations of motion describing evolution of a physical system is proposed. \textcopyright{} 1996 The American Physical Society.
The quantum-mechanical traversal- (tunneling) time problem is reformulated in terms of a square-integrable amplitude distribution (wave function) whose evolution is governed by a generalized Schr\"odinger equation. The method uses an analogy between quantum mechanics and classical Brownian motion. Brownian and quantum traversal-time distributions are evaluated by the direct integration of corresponding equations of motion. The general relation between Feynman and Schr\"odinger quantum mechanics is discussed.
The delay time associated with a scattering process is one of the most important dynamical aspects in quantum mechanics. A common measure of this is the Wigner delay time based on the group velocity description of a wave-packet, which my easily indicate super-luminal or even negative times of interaction that are unacceptable. Many other measures such as dwell times have been proposed, but also suffer from serious deficiencies, particularly for evanescent waves. One important way of realising a timescale that is causally connected to the spatial region of interest has been to utilize the dynamical evolution of extra degrees of freedom called quantum clocks, such as the spin of an electron in an applied magnetic field or coherent decay or growth of light in an absorptive or amplifying medium placed within the region of interest. Here we provide a review of the several approaches developed to answer the basic question - how much time does a quantum particle (or wave) spend in a specified region of space? While a unique answer still evades us, important progress has been made in understanding the timescales and obtaining positive definite times of interaction by noting that all such clocks are affected by spurious scattering concomitant with the very clock potentials, however, weak they be and by eliminating the spurious scattering.
A controversy surrounding the "tunnelling time problem" stems from the seeming inability of quantum mechanics to provide, in the usual way, a definition of the duration a particle is supposed to spend in a given region of space. For this reason, the problem is often approached from an "operational" angle. Typically, one tries to mimic, in a quantum case, an experiment which yields the desired result for a classical particle. One such approach is based on the use of a Larmor clock. We show that the difficulty with applying a non-perturbing Larmor clock in order to "time" a classically forbidden transition arises from the quantum Uncertainty Principle. We also demonstrate that for this reason a Larmor time (in fact, any Larmor time) cannot be interpreted as a physical time interval. We provide a theoretical description of the quantities measured by the clock.
When the solution of the time independent Schrödinger equation is a real function, the velocity of the associated quantum particle is vanishing in Bohmian Mechanics. We can obtain a moving particle if we consider that its velocity can be related not to the phase of the wave function but to its amplitude. By using the rules of standard quantum mechanics, we have constructed a model generalizing the momentum operator and modifying the corresponding Schrödinger equation. We obtain a new nonlinear equation of the form studied by Ermakov and Pinney and we apply the model to the cases of the particle in a box and of a bouncing ball. The new formula of quantum velocity and the corresponding motion of quantum particles have an immediate correspondence with the behavior of the bodies in classical mechanics.
The Larmor clock which was proposed to determine tunneling times of particles with spin through barriers is studied. Tunneling of wave packets is numerically derived from the time-dependent Schr\"odinger-Pauli equation. The Larmor clock often does not proceed unidirectionally in time. The asymptotic readings are well defined. For wide wave packets the asymptotic Larmor times agree with the stationary-state predictions; deviations are seen for narrow wave packets. B\"uttiker's dwell time is investigated and its connection with the Larmor times is confirmed in the case of wave packets. \textcopyright{} 1996 The American Physical Society.