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Physical Review A

Sojourn-time approach to interaction time in quantum scattering: One-dimensional scattering with internal degrees of freedom

Wojciech Jaworski, David M. Wardlaw

Published Nov 1, 1990• 7 citations
Publisher

Abstract

The system studied is a two-level particle whose translational motion is one dimensional and restricted to the half line (0,\ensuremath{\infty}). The internal degrees of freedom of the particle are weakly coupled to the translational motion via a coupling including both adiabatic and nonadiabatic parts. We explore the idea of describing the interaction times of this scattering system in terms of Hermitian sojourn-time operators. Treating the coupling as a perturbation, we show that the first-order perturbation theory can be consistently formulated in terms of appropriate sojourn-time operators. The mean values of these operators give mean sojourn time (dwell time) of the particle in spatial regions or, more generally, in certain subspaces of the state space of the system. The perturbed S operator and also the effect of the perturbation on the change of observables due to the scattering are, in first-order approximation, fully expressible in terms of the sojourn-time operators. In particular, these operators fully determine the first-order approximation to the inelastic scattering probabilities. The relation to the case of scattering on the whole line (-\ensuremath{\infty},\ensuremath{\infty}) is briefly discussed.

DOI
10.1103/physreva.42.5253

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The concept of a time-of-sojourn operator and spreading of wave packets

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The concept of sojourn time and a sojourn time operator, aimed at describing the length of time spent by a quantum mechanical system in a given subspace of states, is investigated. A general rigorous definition to the sojourn time operator is given and some of its properties are studied. In particular, it is shown that the usual Born’s probability interpretation of the associated spectral measure yields strange, if not paradoxical, results, resembling the well-known quantum mechanical Zeno paradox. Also a specific example of a free nonrelativistic particle is considered. Here it is proven that the probability Pt(Ω) of the particle being present in a volume Ω at time t cannot vanish on a set of t having nonzero measure. This implies that the sojourn time in Ω never vanishes, and that zero is never an eigenvalue of the sojourn time operator. It is also shown that for a very general class of sets Ω, including all bounded sets, the sojourn time turns out to be bounded with a bound independent of the initial state of the particle. Correspondingly, the sojourn time operator turns out to be a bounded one.

Journal of Mathematical Physics•Jul 1, 1989•12 citations
The system studied is a two-level particle whose translational motion is one dimensional and restricted to the half line (0,\ensuremath{\infty}). The internal degrees of freedom of the particle are weakly coupled to the translational motion via a coupling including both adiabatic and nonadiabatic parts. We explore the idea of describing the interaction times of this scattering system in terms of Hermitian sojourn-time operators. Treating the coupling as a perturbation, we show that the first-order perturbation theory can be consistently formulated in terms of appropriate sojourn-time operators. The mean values of these operators give mean sojourn time (dwell time) of the particle in spatial regions or, more generally, in certain subspaces of the state space of the system. The perturbed S operator and also the effect of the perturbation on the change of observables due to the scattering are, in first-order approximation, fully expressible in terms of the sojourn-time operators. In particular, these operators fully determine the first-order approximation to the inelastic scattering probabilities. The relation to the case of scattering on the whole line (-\ensuremath{\infty},\ensuremath{\infty}) is briefly discussed.
A general formulation of the sojourn-time operator approach is presented, valid for arbitrary scattering systems and free of restrictions to one-dimensional systems present in the previous work of the authors. We show how first-order perturbation theory can be consistently formulated in terms of appropriate sojourn-time operators. Both time-dependent and time-independent perturbations, including coupling between translation and internal degrees of freedom, are considered. The mean values of these Hermitian sojourn-time operators give the mean sojourn time (dwell time) of the particle in spatial regions or, more generally, in certain subspaces of the state space of the system defined by sets of commuting observables. The perturbed S operator and also the effect of the perturbation on the change of observables due to the scattering are, to a first-order approximation, fully expressible in terms of these operators. We also show that this approach provides a unified treatment of the various idealized one-dimensional systems, which have frequently been employed in the debate on tunneling and interaction times. In particular, we provide some insight into the role of complex times, which appear as combinations of matrix elements of the sojourn-time operators.
We show that a useful concept of the time spent by a quantum-mechanical particle in a given spatial region can be expressed in terms of a Hermitian sojourn time operator. Mean values of this operator give the mean sojourn time (dwell time) of the particle. We show that the sojourn time operators occur in a natural way in first-order perturbation theory for the barrier perturbed by any finite-range potential. The perturbed S operator, and also the effect of the perturbation on the change of observables due to the scattering, are, to a first-order approximation, fully expressible in terms of sojourn time operators. We anticipate that this result can be generalized to the case when the translational motion of the tunneling particle is weakly coupled to additional degrees of freedom. As a specific example of such a system, we explicitly reconsider the Larmor clock, originally proposed as a device measuring interaction times. We show that the change of the spin components, to a first-order approximation, is indeed fully expressible in terms of a sojourn time operator. In particular, when set as a clock, the system measures mean values of the sojourn time operator or real parts of some of its matrix elements.
Numerous studies of one-dimensional scattering systems have failed to provide a satisfactory theory for tunneling times. Although there is some support for the idea that such interaction times should be complex, the physical meaning of complex interaction times has defied explanation. Here we demonstrate an explicit relation between several proposed complex times and a Hermitian sojourn-time operator. We have previously shown that this operator describes the effect of a perturbation of the scattering system on observable quantities.
We discuss some aspects of the time picture of tunneling for open quantum systems described by non-Hermitian (NH) Hamiltonians. The concept of sojourn time for such systems is introduced in the framework of the biorthonormal formalism. Due to the various definitions of probability density in the non-Hermitian case, we get three different sojourn times, two real and one complex. We consider as model of a dissipative NH system the complex, generalized parametric oscillator, for which we derive the exact expressions of the three sojourn times in terms of the Wei-Norman characteristic functions entering the non-unitary evolution operators. The special case of the inverted Caldirola-Kanai oscillator with complex friction parameter is investigated for an initial extended wavepacket. We also discuss the Landau-Zener-like transitions of the NH parametric oscillator, i.e. the dissipative tunneling through a dynamical barrier due to the perturbative effect of the damping.
A ready comparison of the two methods noted in the title is obscured by the very different formalisms that they employ. Our sojourn-time approach [Jaworski and Wardlaw, Phys. Rev. A 45, 292 (1992)] is based on a Hermitian sojourn-time operator in the framework of time-dependent scattering theory whereas Sokolovski and co-workers [Sokolovski and Baskin, Phys. Rev. A 36, 4604 (1987); Sokolovski and Connor, ibid. 42, 6512 (1990)] have developed an approach based on the Feynman path-integral formulation of quantum mechanics. By translating the path-integral approach into the language of wave functions and linear operators, we reveal the similarities of the two approaches and show that the results for the interaction times are closely related, becoming equivalent in the asymptotic limit.
The concept of sojourn time and a sojourn time operator, aimed at describing the length of time spent by a quantum mechanical system in a given subspace of states, is investigated. A general rigorous definition to the sojourn time operator is given and some of its properties are studied. In particular, it is shown that the usual Born’s probability interpretation of the associated spectral measure yields strange, if not paradoxical, results, resembling the well-known quantum mechanical Zeno paradox. Also a specific example of a free nonrelativistic particle is considered. Here it is proven that the probability Pt(Ω) of the particle being present in a volume Ω at time t cannot vanish on a set of t having nonzero measure. This implies that the sojourn time in Ω never vanishes, and that zero is never an eigenvalue of the sojourn time operator. It is also shown that for a very general class of sets Ω, including all bounded sets, the sojourn time turns out to be bounded with a bound independent of the initial state of the particle. Correspondingly, the sojourn time operator turns out to be a bounded one.