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 molecular Schrödinger equation is rewritten in terms of nonunitary equations of motion for the nuclei (or electrons) that depend parametrically on the configuration of an ensemble of generally defined electronic (or nuclear) trajectories. This scheme is exact and does not rely on the tracing out of degrees of freedom. Hence, the use of trajectory-based statistical techniques can be exploited to circumvent the calculation of the computationally demanding Born-Oppenheimer potential-energy surfaces and nonadiabatic coupling elements. The concept of the potential-energy surface is restored by establishing a formal connection with the exact factorization of the full wave function. This connection is used to gain insight from a simplified form of the exact propagation scheme. In order to describe the correlated motion of electrons and nuclei, many strategies have been proposed to transcend the picture where the nuclei evolve on top of a single Born-Oppenheimer potential-energy surface (BOPES) [1]. Using a time-independent basis-set expansion of the electron-nuclear wave function, full quantum studies provide a complete description of nonadiabatic dynamics [2]. The scaling of these methods (even for a time-dependent basis-set expansion [3]) is, however, limiting their use to describe a few degrees of freedom. The so-called direct dynamics techniques attempt to alleviate this problem by calculating the BOPESs on the fly [4]. Of particular interest here are those methods that use information from quantum chemistry or time-dependent density functional theory calculations in the form of forces. Ab initio surface hopping, Ehrenfest dynamics [5], or Gaussian wave packet methods (such as the multiple spawning method) [6] are all able to reproduce the dynamics of some systems of interest [7]. In most of these methods, however, the form of the nuclear wave function is restricted, as they use a local or classical trajectory-based representation of the nuclear wave packet. In addition to the difficulties of including external fields or calculating the nonadiabatic coupling elements (NACs), this introduces the problem of systematically accounting for quantum nuclear effects.In this Letter, we propose an exact propagation scheme aimed at the study of nonadiabatic dynamics in the presence of arbitrary external electromagnetic fields. The coupled electron-nuclear dynamics is separated without tracing out degrees of freedom, which lends itself to a rigorous starting point for systematically including nonadiabatic nuclear effects without relying on the computation of BOPESs and NACs. This work constitutes a multicomponent extension of the conditional formalism proposed in Refs. [8,9]. Further, the propagation scheme presented here generalizes the conditional formalism beyond its original hydrodynamic formulation [8]. This makes it suitable to be coupled with well established electronic structure methods.Throughout this Letter, we use atomic units, and electronic and nuclear coordinates are collectively denoted by r ¼ fr 1 ; …; r N ...

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We report a new theoretical approach to solve adiabatic quantum molecular dynamics halfway between wave function and trajectory-based methods. The evolution of a N-body nuclear wave function moving on a 3N-dimensional Born− Oppenheimer potential-energy hyper-surface is rewritten in terms of single-nuclei wave functions evolving nonunitarily on a 3-dimensional potential-energy surface that depends parametrically on the configuration of an ensemble of generally defined trajectories. The scheme is exact and, together with the use of trajectory-based statistical techniques, can be exploited to circumvent the calculation and storage of many-body quantities (e.g., wave function and potential-energy surface) whose size scales exponentially with the number of nuclear degrees of freedom. As a proof of concept, we present numerical simulations of a 2-dimensional model porphine where switching from concerted to sequential double proton transfer (and back) is induced quantum mechanically. O n the basis of the Born−Huang expansion of the molecular wave function, 1 an exact description of adiabatic molecular dynamics requires the propagation of a nuclear wavepacket on the ground-state Born−Oppenheimer potential-energy surface (gs-BOPES). This propagation scheme is, somehow, computationally doubly prohibitive. Besides the computational burden associated with the propagation of the (many-body) nuclear wave function, the calculation of the gs-BOPES constitutes, per se, a time-independent problem that grows exponentially with the number of electrons and nuclei. In this respect, two main classes of computational methods have emerged depending on whether the knowledge of the gs-BOPES is required in the full configuration space, that is, fullquantum methods, 2,3 or only at certain reduced number of points, namely trajectory-based or direct methods.4,5 While methods for computing the energy of any configuration of nuclei have become quicker and more accurate, full-quantum dynamics calculations still become rapidly unfeasible for large molecules. Alternatively, direct dynamics notably reduce the computational cost of the simulations by avoiding partially, sometimes completely, the calculation of the full gs-BOPES (this can be done, for instance, by the use of reaction-path Hamiltonians 6,7 ). Nuclear quantum effects, however, can be hardly included systematically in this second class of methods. Up to date, only quantum-trajectory methods have the particularity of being able to describe all nuclear quantum effects (just as full-quantum methods) and being on-the-fly simultaneously.8−11 Unfortunately, these methods have serious problems in dealing with the so-called quantum potential, which gathers, by definition, all quantum information on the system. The mathematical structure of the quantum potential depends on the inverse of the quantum probability density, and thus, its manipulation entails serious instability problems. 12−14We report here an exact theoretical approach to solve adiabatic quantum molecular dynamics based o...

We present a protocol for measuring Bohmian -or the mathematically equivalent hydrodynamic -velocities based on an ensemble of two position measurements, defined from a Positive Operator Valued Measure, separated by a finite time interval. The protocol is very accurate and robust as long as the first measurement uncertainty divided by the finite time interval between measurements is much larger than the Bohmian velocity, and the system evolves under flat potential between measurements. The difference between the Bohmian velocity of the unperturbed state and the measured one is predicted to be much smaller than 1% in a large range of parameters. Counterintuitively, the measured velocity is that at the final time and not a time-averaged value between measurements.

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.