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.
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.
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.
Our previous results [Phys. Rev. A 37, 2843 (1988)], based on the concept of sojourn time, for the transmission and reflection time delays in one-dimensional scattering by a potential barrier are compared to the corresponding results of E. H. Hauge, J. P. Falck, and T. A. Fjedly [Phys. Rev. B 36, 4203 (1987)], based on the idea of following the time evolution of the mean position of wave packets. It is shown that the mean-position approach agrees with the more general sojourn-time approach in the limit of well-defined momentum, and that, in general, the two approaches appear to be related in a classical-like manner.
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.
In a paper entitled ’’An Operator Calculus Having Applications in Quantum Electrodynamics’’ [Phys. Rev. 84, 108 (1951)], Feynman gives techniques and theorems for working with ordered operators. We believe that the entire development, because of its direct involvement in the transition from the Heisenberg–Schrödinger to Feynman’s path-integral formulation of quantum mechanics, is of general interest. We aim to make these less appreciated early techniques of Feynman more accessible to the general physicist.
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.
Basically, there are two approaches to the formulation of quantum mechanics, i.e., the operator and the Feynman path integral approach. The main object in quantum mechanics is referred to as the transition amplitude between those states, matrix elements of physical operators and, in particular, the S- matrix in the problem of quantum scattering. On the basis of quantum mechanics, the operator method is closed, well formulated mathematically, and so is an appropriate universal language describing quantum field theory, the quantum physics of condensed states and quantum relativistic statistical physics. The differential equation defining the path of the system in the configuration space is called the equation of motion of the system. The aim of classical as well as quantum mechanics is to predict the time evolution of the system, starting from knowledge of the state of the system at a given point in time.