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The Orthodox kinetic energy has, in fact, two hidden-variable components: one linked to the current (or Bohmian) velocity, and another linked to the osmotic velocity (or quantum potential), and which are respectively identified with phase and amplitude of the wavefunction. Inspired by Bohmian and Stochastic quantum mechanics, we address what happens to each of these two velocity components when the Orthodox kinetic energy thermalizes in closed systems, and how the pertinent weak values yield experimental information about them. We show that, after thermalization, the expectation values of both the (squared) current and osmotic velocities approach the same stationary value, that is, each of the Bohmian kinetic and quantum potential energies approaches half of the Orthodox kinetic energy. Such a `kinetic energy equipartition' is a novel signature of quantum thermalization that can empirically be tested in the laboratory, following a well-defined operational protocol as given by the expectation values of (squared) real and imaginary parts of the local-in-position weak value of the momentum, which are respectively related to the current and osmotic velocities. Thus, the kinetic energy equipartion presented here is independent on any ontological status given to these hidden variables, and it could be used as a novel element to characterize quantum thermalization in the laboratory, beyond the traditional use of expectation values linked to Hermitian operators. Numerical results for the nonequilibrium dynamics of a few-particle harmonic trap under random disorder are presented as illustration. And the advantages in using the center-of-mass frame of reference for dealing with systems with many indistinguishable particles are also discussed., Comment: submitted on 02/May/23

Kirchhoff's Current Law is an essential tool in the design of circuits that operate very quickly, faster than nanoseconds. But Kirchhoff's current is often identified as the flow of particles. The continuity equation or the Maxwell-Ampere law shows that the sum of displacement current $\textbf{plus}$ particle current is conserved by Maxwell's equations and Kirchhoff's law. Kirchoff included the displacement current in the current of his law, from early on. This Kirchhoff current (including the displacement current) does not vary with spatial location in the ionic channels of biology. Electronic circuits switching in nanoseconds are analyzed using the Bohm representation of quantum mechanics including particle and displacement current., Comment: This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible

In this chapter, we will take a trip around several hot-spots where Bohmian mechanics and its capacity to describe the microscopic reality, even in the absence of measurements, can be harnessed as computational tools, in order to help in the prediction of phenomenologically accessible information (also useful for the followers of the Copenhagen theory). As a first example, we will see how a Stochastic Schr\"odinger Equation, when used to compute the reduced density matrix of a non-Markovian open quantum system, necessarily seems to employ the Bohmian concept of a conditional wavefunction. We will see that by dressing these conditional wavefunctions with an interpretation, the Bohmian theory can prove to be a useful tool to build general quantum frameworks, like a high-frequency electron transport model. As a second example, we will introduce how a Copenhagen "observable operator" can be derived from numerical properties of the Bohmian trajectories, which within Bohmian mechanics, are well-defined even for an "unmeasured" system. Most importantly in practice, even if these numbers are given no ontological meaning, not only we will be able to simulate (thus, predict and talk about) them, but we will see that they can be operationally determined in a weak value experiment. Therefore, they will be practical numbers to characterize a quantum system irrespective of the followed quantum theory., Comment: 13 pages, 1 figure, to be published as a Chapter in the book "Physics and the Nature of Reality: Essays in Memory of Detlef D\"urr". Accepted version, integrating comments by referee

We propose a test of macrorealism that exploits the contextuality of two-time correlation functions to escape the so-called "clumsiness loophole" that plagues Leggett-Garg inequalities. The non-contextuality of reduced joint probability distributions is proven to be an unequivocal criterion to guarantee that measurements are carried out in the ideally-weak measurement regime of a class of generalized von Neumann measurements. In this regime, testing the so-called "no-signaling in time" condition allows to uncontextually ascertain whether a property of a given system is macrorealistic or non-macrorealistic. The resulting protocol allows for tests of macrorealism in situations where Leggett-Garg inequalities and ideal negative measurement cannot be used at all.

We discuss the conditions for the classicality of quantum states with a very large number of identical particles. By treating the center of mass as a Bohmian particle, we show that it follows a classical trajectory when the distribution of the Bohmian positions in just one experiment is always equal to the marginal distribution of the quantum state in physical space. This result can also be interpreted as a unique-experiment generalization of the well-known Ehrenfest theorem. We also demonstrate that the classical trajectory of the center of mass is fully compatible with a conditional wave function solution of a classical non-linear Schr\"odinger equation. Our work shows clear evidence for a quantum-classical inter-theory unification and opens new possibilities for practical quantum computations with decoherence., Comment: 29 pages, 5 figures

After the seminal work of R. Landauer in 1957 relating the electrical resistance of a conductor to its scattering properties, much progress has been made in our ability to predict the performance of electron devices in the DC (stationary) regime. Computational tools to describe their dynamical behavior (including the AC, transient and noise performance), however, are far from being as trustworthy as would be desired by the electronic industry. While there is no fundamental limitation to correctly modeling the high-frequency quantum transport and its fluctuations, certainly more careful attention must be paid to delicate issues such as overall charge neutrality, total current conservation, or the back action of the measuring apparatus. In this review, we will show how the core ideas behind the Bohmian formulation of quantum mechanics can be exploited to design an efficient Monte Carlo algorithm that provides a quantitative description of electron transport in open quantum systems. By making the most of trajectory-based and wave function methods, the BITLLES simulator, a free software developed by the authors, extends the capabilities that the semi-classical Monte Carlo simulation technique has offered for decades (DC, AC, noise, transients) to the quantum regime.

In this paper, the classical Schr��dinger equation, which allows the study of classical dynamics in terms of wave functions, is analyzed theoretically and numerically. First, departing from classical (Newtonian) mechanics, and assuming an additional single-valued condition for the Hamilton's principal function, the classical Schr��dinger equation is obtained. This additional assumption implies inherent non-classical features on the description of the dynamics obtained from the classical Schr��dinger equation: the trajectories do not cross in the configuration space. Second, departing from Bohmian mechanics and invoking the quantum-to-classical transition, the classical Schr��dinger equation is obtained in a natural way for the center of mass of a quantum system with a large number of identical particles. This quantum development imposes the condition of dealing with a narrow wave packet, which implicitly avoids the non-classical features mentioned above. We illustrate all the above points with numerical simulations of the classical and quantum Schr��dinger equations for different systems., 11 pages, 7 figures

This chapter provides a comprehensive overview of the Bohmian formulation of quantum mechanics. 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 Schrodinger 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., Comment: 87 pages. This paper includes the two prefaces, the introduction and the first chapter of the second edition of the book: Applied Bohmian Mechanics: From Nanoscale Systems to Cosmology (Print ISBN: 9789814800105). The whole book, with nine additional chapters, has been published by Pan Stanford editorial and it can be ordered and it can be ordered at: sales@panstanford.com

We discuss and solve the transport without transit quantum paradox recently introduced in the context of the adiabatic transport of a single particle or a Bose--Einstein condensate between the two extreme traps of a triple-well potential. To this aim, we address the corresponding quantum dynamics in terms of Bohmian trajectories and show that transport always implies transit through the middle well, in full agreement with the quantum continuity equation. This adiabatic quantum transport presents a very counterintuitive effect: by slowing down the total time duration of the transport process, ultra-high Bohmian velocities are achieved such that, in the limit of perfect adiabaticity, relativistic corrections are needed to properly address the transfer process while avoiding superluminal matter wave propagation., Comment: 4 pages, 3 figures