R. Sala

Transmission and reflection tunneling times

Abstract Different separations of the dwell time into real reflection and transmission partial times for tunneling involving wave packet scattering are proposed. The formalism is valid in both quantum mechanics and classical statistical mechanics. Numerical examples are provided.

Wigner trajectories and Liouville’s theorem

It is found that, in general, Wigner trajectories satisfy Liouville’s theorem only locally, i.e., for restricted phase space and time domains. This fact limits their possible applications. Examples are provided to visualize the process of creation and destruction of Wigner trajectories. It is argued that Weyl transforms of Heisenberg operators are, however, viable alternatives to Wigner trajectories, even though they do not satisfy Liouville’s theorem either.

The time of arrival concept in quantum mechanics

The concept and the formalization of the arrival time in quantum mechanics are discussed. Different approaches based on trajectories, quantization rules, time operators, phase-space techniques, renewal equations or operational procedures are reviewed or proposed. Open questions and loose ends are pointed out.

Average local values and local variances in quantum mechanics

Several definitions for the average local value and local variance of a quantum observable are examined and compared with their classical counterparts. An explicit way to construct an infinite number of these quantities is provided. It is found that different classical conditions may be satisfied by different definitions, but none of the quantum definitions examined is entirely consistent with all classical requirements.

COMPOSITE ABSORBING POTENTIALS

The multiple scattering interferences due to the addition of several contiguous potential units are used to construct composite absorbing potentials that absorb at an arbitrary set of incident momenta or for a broad momentum interval.

Comparison of positive flux operators for transition state theory using a solvable model

Several quantum operators representing ‘‘positive flux’’ are compared for the square barrier by examining their ability to reproduce the exact transmittance when traced with the exact microcanonical density operator. They are obtained by means of the ‘‘Weyl rule,’’ the ‘‘Rivier rule,’’ by symmetrizing the product of ‘‘flux’’ and ‘‘positive momentum projection’’ operators, and by a variational technique. Explicit expressions are given for all cases.

Phase space formalisms of quantum mechanics with singular kernel

Abstract The equivalence of the Rivier-Margenau-Hill and Born-Jordan-Shankara phase space formalisms to the conventional operator approach of quantum mechanics is demonstrated. It is shown that in spite of the presence of singular kernels mappings relating phase space functions and operators are possible both backward and forward.

Transmission, Reflection, and Interference Contributions to the Tunnelling Dwell Time

Expressions for the transmission, reflection, and interference contributions to the dwell time for wave packet scattering are obtained by means of scattering theory projector operators that provide a unifying framework for various dwell time decompositions. These different dwell time resolutions are illustrated with the aid of a formal analogy between the two-level system and one-dimensional scattering. The complex times and Larmor times proposed by other authors are found as particular cases of the theory. Numerical calculations for a rectangular barrier are provided.

Wigner function for the square barrier

Abstract The Wigner function for the scattering of an incident plane wave with definite momentum impinging on a square potential barrier is obtained explicitly. The associated Wigner trajectories are examined and compared with Wigner trajectories corresponding to other model potentials and boundary conditions. Conditions of applicability and non-classical features of Wigner trajectories are discussed.

Equivalence between tunnelling times based on: (a) absorption probabilities, (b) the Larmor clock, and (c) scattering projectors

The assumption of analyticity of the transmission and reflection probability amplitudes as functions of a complex potential is shown to be justified. As a consequence the reflection and transmission times based on absorption probabilities are shown to be equal to the corresponding times derived from the local Larmor precession in the plane perpendicular to the magnetic field. An additional new interpretaton of these times is provided by means of a more general scattering theory projector formalism. The relation to the phase times is discussed.