Complex absorbing potentials

Characteristic Times in One-Dimensional Scattering.- The Time-Energy Uncertainty Relation.- Jump Time and Passage Time: The Duration ofs a Quantum Transition.- Bohm Trajectory Approach to Timing Electrons.- Decoherent Histories for Space-Time Domains.- Quantum Traversal Time, Path Integrals and "Superluminal" Tunnelling.- Quantum Clocks and Stopwatches.- The Local Larmor Clock, Partial Densities of States, and Mesoscopic Physics.- "Standard" Quantum-Mechanical Approach to Times of Arrival.- Experimental Issues in Quantum-Mechanical Time Measurement.- Microwave Experiments on Tunneling Time.- The Two-State Vector Formalism: An Updated Review.

Time in quantum mechanics

Arrival time in quantum mechanics

If frequent measurements ascertain whether a quantum system is still in its initial state, transitions to other states are hindered and the quantum Zeno effect takes place. However, in its broader formulation, the quantum Zeno effect does not necessarily freeze everything. In contrast, for frequent projections onto a multidimensional subspace, the system can evolve away from its initial state, although it remains in the subspace defined by the measurement. The continuing time evolution within the projected 'quantum Zeno subspace' is called 'quantum Zeno dynamics': for instance, if the measurements ascertain whether a quantum particle is in a given spatial region, the evolution is unitary and the generator of the Zeno dynamics is the Hamiltonian with hard-wall (Dirichlet) boundary conditions. We discuss the physical and mathematical aspects of this evolution, highlighting the open mathematical problems. We then analyze some alternative strategies to obtain a Zeno dynamics and show that they are physically equivalent.

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Quantum Zeno dynamics: mathematical and physical aspects

The progress of the factorization method since the 1935 work of Dirac is briefly reviewed. Though linked with older mathematical theories the factorization seems an autonomous 'driving force', offering substantial support to the present day Darboux and Backlund approaches.

http://homepage.cem.itesm.mx/fdelgado/ciencia/cadi/ref06.pdf

Factorization: little or great algorithm?

In a recent paper Grot, Rovelli, and Tate (GRT) [Phys. Rev. A 54, 4676 (1996)] derived an expression for the probability distribution $\ensuremath{\pi}(T;X)$ of intrinsic arrival times $T(X)$ at position $x=X$ for a quantum particle with initial wave function $\ensuremath{\psi}(x,t=0)$ freely evolving in one dimension. This was done by quantizing the classical expression for the time of arrival of a free particle at $X,$ assuming a particular choice of operator ordering, and then regulating the resulting time of arrival operator. For the special case of a minimum-uncertainty-product wave packet at $t=0$ with average wave number $〈k〉$ and variance $\ensuremath{\Delta}k$ they showed that their analytical expression for $\ensuremath{\pi}(T;X)$ agreed with the probability current density $J(x=X,t=T)$ only to terms of order $\ensuremath{\Delta}k/〈k〉.$ They dismissed the probability current density as a viable candidate for the exact arrival time distribution on the grounds that it can sometimes be negative. This fact is not a problem within Bohmian mechanics where the arrival time distribution for a particle, either free or in the presence of a potential, is rigorously given by $|J(X,T)|$ (suitably normalized) [W. R. McKinnon and C. R. Leavens, Phys. Rev. A 51, 2748 (1995); C. R. Leavens, Phys. Lett. A 178, 27 (1993); M. Daumer et al., in On Three Levels: The Mathematical Physics of Micro-, Meso-, and Macro-Approaches to Physics, edited by M. Fannes et al. (Plenum, New York, 1994); M. Daumer, in Bohmian Mechanics and Quantum Theory: An Appraisal, edited by J. T. Cushing et al. (Kluwer Academic, Dordrecht, 1996)]. The two theories are compared in this paper and a case presented for which the results could not differ more: According to GRT's theory, every particle in the ensemble reaches a point $x=X,$ where $\ensuremath{\psi}(x,t)$ and $J(x,t)$ are both zero for all $t,$ while no particle ever reaches $X$ according to the theory based on Bohmian mechanics. Some possible implications are discussed.

Time of arrival in quantum and Bohmian mechanics

Arrival time distributions and perfect absorption in classical and quantum mechanics

Arrival time distributions

In Bohm's causal or trajectory interpretation of quantum mechanics, it is straightforward to determine, from a sufficiently large number of calculated particle trajectories, probability distributions for time delays caused by a potential barrier, or for transmission times through a barrier. We show that these distributions can be calculated directly and more efficiently from probability currents, without the calculation of Bohm trajectories as an intermediate step. The ideas are illustrated for Gaussian wave packets incident on a square potential barrier and used to explain why average causal delay times differ from the average delay times calculated in other approaches.

C.R. Leavens

Papers

Arrival time in quantum mechanics

Time of arrival in quantum and Bohmian mechanics

Arrival time distributions and perfect absorption in classical and quantum mechanics

Arrival time distributions

Distributions of delay times and transmission times in Bohm's causal interpretation of quantum mechanics