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Physical Review Letters

New Physical Interpretation for Time in Scattering Theory

Eli Pollak, William Hughes Miller

Published Jul 9, 1984• 153 citations
Publisher

Abstract

We show that the collision time may be interpreted as the time average of a flux-flux correlation function. This interpretation leads quantum mechanically to a complex time whose real part is identical to the usual definition as provided by F. T. Smith. The imaginary part is identical, in the semiclassical limit, to the imaginary time associated with tunneling.

DOI
10.1103/physrevlett.53.115

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•Jul 31, 2022•0 citations
We show that the collision time may be interpreted as the time average of a flux-flux correlation function. This interpretation leads quantum mechanically to a complex time whose real part is identical to the usual definition as provided by F. T. Smith. The imaginary part is identical, in the semiclassical limit, to the imaginary time associated with tunneling.
The concept of time in quantal tunneling processes is reexamined. We find that the Wigner–Eisenbud definition of real time and the definition of imaginary time may be understood in terms of stationary phase analysis in complex time of the microcanonical flux–flux correlation function. This analysis explains why the real time should not be used to justify adiabatic approximations for perpendicular degrees of freedom at tunneling energies. A semiclassical analysis shows that, as suggested by Buettiker and Landauer, the imaginary time should be used to determine the validity of the adiabatic approximation for tunneling processes. Numerical examples for the hydrogen exchange reaction are provided. The implications on adiabatic and sudden approximations in reactive scattering are discussed. A theory unifying the two approaches is outlined.
We present an interpretation of traversal time results in Josephson junctions, based on a stochastic model of tunneling processes which include dissipative effects. With the use of semiclassical analysis, the experiments performed up until now have supplied a time duration of the order of 102ps{10}^{2}\mathrm{ps}102ps (a remarkably long time, even considering the macroscopic nature of the systems under examination). According to our interpretation, this duration is only the imaginary part of a complex quantity, the real part of which (presumably the true physical duration) lies in the range of a few picoseconds.
In this paper, the wave function is interpreted as space-time: under any representation, the real part of ‎the wave function ‎represents space and the imaginary part represents time. A wave function represents ‎the quantum entanglement (correlation) ‎between time and space. The unitary evolution process is the ‎mutual transformation of space-time. The imaginary time ‎indicates that it can entangle with space in a ‎nonlocalized way and dominate the integrity (unification) of a quantum, which ‎is the essence of ‎quantum entanglement.‎
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.
In this paper, the wave function is interpreted as space-time: under any representation, the real part of the wave function represents space and the imaginary part represents time. A wave function represents the quantum entanglement (correlation) between time and space. The unitary evolution process is the mutual transformation of space-time. The imaginary time indicates that it can entangle with space in a non localized way and dominate the integrity (unification) of a quantum, which is the essence of quantum entanglement.