Motivated by a recent approach to solve quantum dynamics with full Coulomb correlations [X. Oriols, Phys. Rev. Lett.98 (2007) 066803], we present here an extension of the Ramo–Shockley–Pellegrini theorem for quantum systems to compute the total (conduction plus displacement) current in terms of quantum (Bohmian) trajectories. By way of test, we derive an extension of the Ramo-Shockley-Pellegrini theorem using standard quantum mechanics and we compare it to our former result. As expected, both formulations give identical results, however we emphasize the numerical viability of computing self-consistently the total current by means of quantum trajectories in front of the difficulties to do it in terms of standard quantum mechanics.
Motivated by a recent approach to solve quantum dynamics with full Coulomb correlations [X. Oriols, Phys. Rev. Lett.98 (2007) 066803], we present here an extension of the Ramo–Shockley–Pellegrini theorem for quantum systems to compute the total (conduction plus displacement) current in terms of quantum (Bohmian) trajectories. By way of test, we derive an extension of the Ramo-Shockley-Pellegrini theorem using standard quantum mechanics and we compare it to our former result. As expected, both formulations give identical results, however we emphasize the numerical viability of computing self-consistently the total current by means of quantum trajectories in front of the difficulties to do it in terms of standard quantum mechanics.
A recent proposal for modeling time-dependent quantum electron transport with Coulomb and exchange correlations using quantum (Bohm) trajectories (Oriols 2007 Phys. Rev. Lett. 98 066803) is extended towards the computation of the total (particle plus displacement) current in mesoscopic devices. In particular, two different methods for the practical computation of the total current are compared. The first method computes the particle and the displacement currents from the rate of Bohm particles crossing a particular surface and the time-dependent variations of the electric field there. The second method uses the Ramo–Shockley theorem to compute the total current on that surface from the knowledge of the Bohm particle dynamics in a 3D volume and the time-dependent variations of the electric field on the boundaries of that volume. From a computational point of view, it is shown that both methods achieve local current conservation, but the second is preferred because it is free from 'spurious' peaks. A numerical example, a Bohm trajectory crossing a double-barrier tunneling structure, is presented, supporting the conclusions.
The extension of the Ramo–Schockley–Pellegrini theorem for quantum systems allows to define a positive-operator valued measure (POVM) for the total conduction plus displacement electrical current. The resulting current operator is characterized by two parameters, viz. the width of the associated Gaussian functions and the lapse of time between consecutive measurements. For large Gaussian dispersions and small time intervals, the operator obeys to a continuous weak-measurement scheme. Contrarily, in the limit of very narrow Gaussian widths and a single-shot measurement, the operator corresponds to a standard von Neumann (projective) measurement. We have implemented the resulting measurement protocol into a quantum electron transport simulator. Numerical results for a resonant tunneling diode show the great sensibility of current–voltage characteristics to different parameter configurations of the total current operator.
Following Oriols (2007 Phys. Rev. Lett. 98 066803), an algorithm to deal with the exchange interaction in non-separable quantum systems is presented. The algorithm can be applied to fermions or bosons and, by construction, it exactly ensures that any observable is totally independent of the interchange of particles. It is based on the use of conditional Bohmian wave functions which are solutions of single-particle pseudo-Schrödinger equations. The exchange symmetry is directly defined by demanding symmetry properties of the quantum trajectories in the configuration space with a universal algorithm, rather than through a particular exchange-correlation functional introduced into the single-particle pseudo-Schrödinger equation. It requires the computation of N(2) conditional wave functions to deal with N identical particles. For separable Hamiltonians, the algorithm reduces to the standard Slater determinant for fermions (or permanent for bosons). A numerical test for a two-particle system, where exact solutions for non-separable Hamiltonians are computationally accessible, is presented. The numerical viability of the algorithm for quantum electron transport (in a far-from-equilibrium time-dependent open system) is demonstrated by computing the current and fluctuations in a nano-resistor, with exchange and Coulomb interactions among electrons.
This chapter provides a comprehensive overview of the Bohmian formulation of quantummechanics. It starts with a historical review of the difficulties found by Louis de Broglie, David Bohm, and John S. Bell to convince the scientific community about the validity and utility of Bohmian mechanics. Then, a formal explanation of Bohmian mechanics for nonrelativistic, single-particle quantum systems is presented. The generalization to many-particle systems, where the exchange interaction and the spin play an important role, is also presented. After that, the measurement process in Bohmian mechanics is discussed. It is emphasized that Bohmian mechanics exactly reproduces the mean value and temporal and spatial correlations obtained from the standard, that is the Copenhagen or orthodox, formulation. The ontological characteristics of Bohmian mechanics provide a description of measurements as another type of interaction without the need for introducing the wave function collapse. Several solved problems are presented at the end of the chapter, giving additional mathematical support to some particular issues. A detailed description of computational algorithms to obtain Bohmian trajectories from the numerical solution of the Schro¨dinger or the Hamilton-Jacobi equations are presented in an appendix. The motivation of this chapter is twofold: first, as a didactic introduction to Bohmian formalism, which is used in the subsequent chapters, and second, as a self-contained summary for any newcomer interested in using Bohmian mechanics in his or her daily research activity.
In this chapter, a glimpse of the way how the concept of quantum trajectories has been used in different areas of quantum chemistry and chemical physics is provided. Putting aside ontological issues, Bohmian mechanics has been exploited within these fields with practical purposes, both analytical and computational. Analytically, although quantum trajectories provide the same information at a predictive level as standard quantum mechanics, they have the advantage that one can follow and describe the evolution of quantum systems by means of streamlines, whose results are less vague and more intuitive than standard wave packet simulations. On the other hand, computationally, Bohmian mechanics has been the source of a number of numerical algorithms devised as an alternative to other methods based on the wave function or the probability density. This will be seen throughout the analysis and discussion of different applications, which go from electronic structure to chemical dynamics, thus covering essentially the research spectrum of quantum chemistry and chemical physics. Furthermore, in the form of a final summary, we present a unifying scheme based on minimum principles, which allows us to connect electronic structurewith classical and quantum (Bohmian) chemical dynamics.
In a recent paper [B. Poirier, Chem. Phys. 370, 4 (2010)], a formulation of quantum mechanics was presented for which the usual wavefunction and Schrödinger equation are replaced with an ensemble of real-valued trajectories satisfying a principle of least action. It was found that the resultant quantum trajectories are those of Bohmian mechanics. In this paper, analogous ideas are applied to Bohmian Mechanics with Complex Action (BOMCA). The standard BOMCA trajectories as previously defined are found not to satisfy an action principle. However, an alternate set of complex equations of motion is derived that does exhibit this desirable property, and an approximate numerical implementation is presented. Exact analytical results are also presented, for Gaussian wavepacket propagation under quadratic potentials.