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Physical Review A

Path integrals and equations of motion for the traversal-time distributions in classical diffusion and quantum mechanics

D. Sokolovski

Published Jul 1, 1995• 34 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.52.r5

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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.
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.
In Feynman's space-time approach to nonrelativistic quantum mechanics, the wave function may be considered to be the sum of path integrals over Brownian-motion trajectories. This formalism is entirely equivalent to the Schr\"odinger-equation treatment with its intrinsic indeterminacy. Pursuing the analogy with Brownian motion further, one may introduce a hidden variable corresponding to particle velocity in order to represent the wave function as a sum of path integrals taken over phase-space trajectories. By means of a hidden parameter corresponding to the time scale of fictitious interactions of the particle with the vacuum, one can define generalized propagators which over very short times produce wave functions which are localizable in phase space and hence are deterministic. For long times the conventional Feynman formalism is recovered. This short-time causality does not violate the von Neumann injunction against hidden variables since the new velocity variable turns out to be non-Hermitian over the long time scales normally considered. In contrast to the models proposed by Bohm and others, the present Brownian model is linear and preserves the usual statistical interpretation of the wave function for sufficiently long times.
The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to noncommuting operators. Canonically conjugate observables have to be noncommuting operators in order to satisfy the Heisenberg uncertainty principle. The dynamics of a quantum mechanical system is given by the time-dependent Schrödinger equation. In the early 1940s Feynman realized that it would be possible to construct a quantum model using the classical Lagrangian approach (a method closely related to the action integral). Dirac had earlier pointed out that eit times the Lagrangian was analogous to a transformation function for the wave function in which the wave function at one instant could be related to the wave function at a next instant in a time interval of t by multiplying with such an exponential function. Feynman found the exponential phase factor to be the action S which is the time integral of the Lagrangian. Further, he noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. He developed a way for describing and evaluating the integration using the idea of integral over all paths. In Feynman’s approach, the particles are described using amplitudes calculated along the paths they may or may not follow. These amplitudes behave like waves, their phase changes as the quantum system moves along the path and the amplitudes for all paths superpose and interfere. This interference between all paths is called as sum-over-histories and the resulting amplitude is connected in the same way that the amplitude of a wave function is linked to probability as magnitude-squared.
The quantum traversal time ${t}_{\ensuremath{\Omega}}$ is obtained as a matrix element of the classical functional in Feynman path-integral technique. The properties of ${t}_{\ensuremath{\Omega}}$ and its relation to the observables in scattering experiments are investigated. The general expression for the tunneling time is found in the one-dimensional case. Various traversal times of previous works are shown to be simply related to ${t}_{\ensuremath{\Omega}}$ as Re${t}_{\ensuremath{\Omega}}$, Im${t}_{\ensuremath{\Omega}}$, or \ensuremath{\Vert}${t}_{\ensuremath{\Omega}}$\ensuremath{\Vert}. Other possible constructions of time parameters describing the motion of a quantum particle are discussed.
The Feynman path integral formulation of quantum mechanics is a path integral representation for a propagator or probability amplitude in going between two points in space-time. The wave function is expressed in terms of an integral equation from which the Schrodinger equation can be derived. On taking the limit h — 0, the method of stationary phase can be applied and Newton's second law of motion is obtained. Also, the condition the phase vanishes leads to the Hamilton - Jacobi equation. The secondary objective of this paper is to study ways of relating quantum mechanics and classical mechanics. The Ehrenfest theorem is applied to a particle in an electromagnetic field. Expressions are found which are the hermitian Lorentz force operator, the hermitian torque operator, and the hermitian power operator.