It is proved that many-particle Bohm trajectories can be computed from single-particle time-dependent Schrödinger equations. From this result, a practical algorithm for the computation of transport properties of many-electron systems with exchange and Coulomb correlations is derived. As a test, two-particle Bohm trajectories in a tunneling scenario are compared to exact results with an excellent agreement. The algorithm opens the path for implementing a many-particle quantum transport (Monte Carlo) simulator, beyond the Fermi liquid paradigm. DOI: 10.1103/PhysRevLett.98.066803 PACS numbers: 73.23.ÿb, 02.70.Tt, 34.10.+x From a computational point of view, the direct solution of the many-particle Schrödinger equation is inaccessible for more than very few electrons. This issue is at the heart of almost all the unsolved problems in quantum transport. The standard solution to overcome this computational barrier is assuming noninteracting (Fermi liquid) electrons and decoupling the studies of transport from those of the electronic structure (via the effective electron mass) [1]. Nowadays, ab initio many-particle quantum transport approaches, based on the density functional theory (DFT) [2], are being developed [3] to surpass the previous approximations.In this Letter, we present an alternative approach, beyond the Fermi liquid paradigm, to study many-particle quantum transport using Bohm trajectories [4,5]. Bohmian mechanics was originally presented as an interpretative tool, and it generated an intense debate about the ''reality'' of the trajectories [6]. Here, this issue becomes irrelevant because Bohm trajectories (similarly to Feynman paths) are used to reproduce the probabilistic results of standard quantum mechanics. Following this point of view, Bohmian mechanics has recently undergone a revival to develop new quantum computational algorithms [7].We study a system of N (spinless) electrons described by a (first-quantization) many-particle wave-function, x; t, solution of the Schrödinger equation:wherex fx 1 ; x 2 ; . . . ; x N g fx a ;x b g is the vector of the electron positions. For simplicity, we consider a 1D solidstate system where the lattice-electron interaction is included into the electron effective mass, m. The potential energy Ux; t takes into account the Coulomb interaction among all electrons and the role of an external battery. First, we summarize the basic many-particle de BroglieBohm development [4,5,7,8]. Equation (1) can be split into two (real) equations when the wave function is written in polar form, x; t Rx; t expiSx; t=@. The real part leads to the many-particle quantum Hamilton-Jacobi Equations (2) The vectorxt fx 1 t; . . . ; x N tg fx a t;x b tg contains the N Bohm trajectories. We omit the dependence of each trajectory on its particular initial position, x a t o .On the other hand, the imaginary part of Eq. (1) leads to a continuity equation:

Abstract. Bohmian mechanics provides an explanation of quantum phenomena in terms of point-like particles guided by wave functions. This review focuses on the use of nonrelativistic Bohmian mechanics to address practical problems, rather than on its interpretation. Although the Bohmian and standard quantum theories have different formalisms, both give exactly the same predictions for all phenomena. Fifteen years ago, the quantum chemistry community began to study the practical usefulness of Bohmian mechanics. Since then, the scientific community has mainly applied it to study the (unitary) evolution of single-particle wave functions, either by developing efficient quantum trajectory algorithms or by providing a trajectorybased explanation of complicated quantum phenomena. Here we present a large list of examples showing how the Bohmian formalism provides a useful solution in different forefront research fields for this kind of problems (where the Bohmian and the quantum hydrodynamic formalisms coincide). In addition, this work also emphasizes that the Bohmian formalism can be a useful tool in other types of (nonunitary and nonlinear) quantum problems where the influence of the environment or the nonsimulated degrees of freedom are relevant. This review contains also examples on the use of the Bohmian formalism for the many-body problem, decoherence and measurement processes. The ability of the Bohmian formalism to analyze this last type of problems for (open) quantum systems remains mainly unexplored by the scientific community. The authors of this review are convinced that the final status of the Bohmian theory among the scientific community will be greatly influenced by its potential success in those types of problems that present nonunitary and/or nonlinear quantum evolutions. A brief introduction of the Bohmian formalism and some of its extensions are presented in the last part of this review.

A many-particle Hamiltonian for a set of particles with Coulomb interaction inside an open system is described without any perturbative or mean-field approximation. The boundary conditions of the Hamiltonian on the borders of the open system ͓in the real three-dimensional ͑3D͒ space representation͔ are discussed in detail to include the Coulomb interaction between particles inside and outside of the open system. The manyparticle Hamiltonian provides the same electrostatic description obtained from the image-charge method, but it has the fundamental advantage that it can be directly implemented into realistic ͑classical or quantum͒ electron device simulators via a 3D Poisson solver. Classically, the solution of this many-particle Hamiltonian is obtained via a coupled system of Newton-type equations with a different electric field for each particle. The quantum-mechanical solution of this many-particle Hamiltonian is achieved using the quantum ͑Bohm͒ trajectory algorithm ͓X. Oriols, Phys. Rev. Lett. 98, 066803 ͑2007͔͒. The computational viability of the manyparticle algorithms to build powerful nanoscale device simulators is explicitly demonstrated for a ͑classical͒ double-gate field-effect transistor and a ͑quantum͒ resonant tunneling diode. The numerical results are compared with those computed from time-dependent mean-field algorithms showing important quantitative differences.

The ontology of Bohmian mechanics includes both the universal wave function (living in 3N-dimensional configuration space) and particles (living in ordinary 3-dimensional physical space). Proposals for understanding the physical significance of the wave function in this theory have included the idea of regarding it as a physicallyreal field in its 3N-dimensional space, as well as the idea of regarding it as a law of nature. Here we introduce and explore a third possibility in which the configuration space wave function is simply eliminated-replaced by a set of single-particle pilotwave fields living in ordinary physical space. Such a re-formulation of the Bohmian pilot-wave theory can exactly reproduce the statistical predictions of ordinary quantum theory. But this comes at the rather high ontological price of introducing an infinite network of interacting potential fields (living in 3-dimensional space) which influence the particles' motion through the pilot-wave fields. We thus introduce an alternative approach which aims at achieving empirical adequacy (like that enjoyed by GRW type theories) with a more modest ontological complexity, and provide some preliminary evidence for optimism regarding the (once popular but prematurely-abandoned) program of trying to replace the (philosophically puzzling) configuration space wave function with a (totally unproblematic) set of fields in ordinary physical space.

Electricity plays a special role in our lives and life. Equations of electron dynamics are nearly exact and apply from nuclear particles to stars. These Maxwell equations include a special term the displacement current (of vacuum). Displacement current allows electrical signals to propagate through space. Displacement current guarantees that current is exactly conserved from inside atoms to between stars, as long as current is defined as Maxwell did, as the entire source of the curl of the magnetic field. We show how the Bohm formulation of quantum mechanics allows easy definition of current. We show how conservation of current can be derived without mention of the polarization or dielectric properties of matter. Matter does not behave the way physicists of the 1800's thought it does with a single dielectric constant, a real positive number independent of everything. Charge moves in enormously complicated ways that cannot be described in that way, when studied on time scales important today for electronic technology and molecular biology. Life occurs in ionic solutions in which charge moves in response to forces not mentioned or described in the Maxwell equations, like convection and diffusion. Classical derivations of conservation of current involve classical treatments of dielectrics and polarization in nearly every textbook. Because real dielectrics do not behave in a classical way, classical derivations of conservation of current are often distrusted or even ignored. We show that current is conserved exactly in any material no matter how complex the dielectric, polarization or conduction currents are. We believe models, simulations, and computations should conserve current on all scales, as accurately as possible, because physics conserves current that way. We believe models will be much more successful if they conserve current at every level of resolution, the way physics does.Comment: Version 4 slight reformattin

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Weak values allow the measurement of observables associated with noncommuting operators. Up to now, position-momentum weak values have been mainly developed for (relativistic) photons. In this Letter, a proposal for the measurement of such weak values in typical electronic devices is presented. Inspired by the Ramo-Shockley-Pellegrini theorem that provides a relation between current and electron velocity, it is shown that the displacement current measured in multiterminal configurations can provide either a weak measurement of the momentum or strong measurement of position. This proposal opens new opportunities for fundamental and applied physics with state-ofthe-art electronic technology. As an example, a setup for the measurement of the Bohmian velocity of (nonrelativistic) electrons is presented and tested with numerical experiments.Introduction.-Nowadays, there is a rapidly growing interest in weak measurements and weak values [1-4], both from fundamental and applied points of view. Since weak values (a weak measurement postselected by a strong measurement) provide information on incompatible observables associated with noncommuting operators, relevant topics of quantum mechanics, such as the tunneling times [5], Hardy's paradox [6,7], Leggett-Garg inequalities [8,9], and quantum amplification [10][11][12], are being revisited. Especially attractive is the simultaneous measurement of position and momentum: a set of weak measurements of position postselected by a strong measurement of momentum is proportional to the wave function of the system [13,14], while a weak measurement of momentum postselected by a strong measurement of position gives the local velocity of a quantum particle [15,16].Most experimental techniques for weak values are developed for photons, whose technology is not easily transferable to industry based on electronics. The few proposals dealing with weak measurements in solid-state systems [17-22] use particle current measurement (i.e., electron charge detection). Instead, we propose measuring displacement current (i.e. time dependent variations of the electric field) to get information on the position and momentum of a quantum state. Similar to Landauer's proposal [23] which demonstrates that the measured dc current provides information of the quantum transmission coefficient, here, we show that the weak measurement of the ac current flowing through a (properly prepared multiterminal) electron device provides information on the whole quantum state. This new proposal opens original routes to study, both, fundamental physics and quantum engineering.As an example of the potentialities of our proposal, inspired by the old classical works of Shockley and Ramo [24,25], we discuss the measurement of the local (Bohmian) velocity (i.e. the current density divided by the modulus of the wave function) for an electron. Such velocity is obtained from a weak value constructed from two measurements of the displacement current on two different metallic surfaces belonging to a multiterminal

Without access to the full quantum state, modeling dissipation in an open system requires approximations. The physical soundness of such approximations relies on using realistic microscopic models of dissipation that satisfy completely positive dynamical maps. Here we present an approach based on the use of the Bohmian conditional wave function that, by construction, ensures a completely positive dynamical map for either Markovian or non-Markovian scenarios, while allowing the implementation of realistic dissipation sources. Our approach is applied to compute the current-voltage characteristic of a resonant tunneling device with a parabolic-band structure, including electron-lattice interactions. A stochastic Schrödinger equation is solved for the conditional wave function of each simulated electron. We also extend our approach to (graphene-like) materials with a linear band-structure using Bohmian conditional spinors for a stochastic Dirac equation.