Commonly held opinion is that particle trajectory descriptions are incompatible with quantum mechanics. Louis de Broglie (1926) first proposed a way to include trajectories in quantum mechanics, but the idea was abandoned until David Bohm (1952) re-invented and improved the theory. Bohm interprets the particle trajectories as physically real; for example, an electron actually is a particle moving on a well defined trajectory with a position and momentum at all times. By design, Bohm's trajectories never make predictions that differ from standard quantum mechanics, and their existence cannot be experimentally verified. Three new methods to obtain Bohm's particle trajectories are presented. The methods are non-dynamical, and utilize none of Bohm's equations of motion; in fact, two of the methods have no equations for a particle's trajectory. Instead, all three methods use only the evolving probability density ρ=ψ*ψ to extract the trajectories. The first two methods rest upon probability conservation and density sampling, while the third method employs the informational or geometrical construction of centroidal Voronoi tessellations. In one-dimension all three methods are proved to be equivalent to Bohm's particle trajectories. For higher dimensional configuration spaces, the first two methods can be used in limited situations, but the last method can be applied in all cases. Typically, the resulting higher dimensional non-dynamical trajectories are also identical to Bohm. Together the three methods point to a new interpretation of Bohm's particle trajectories, namely, the Bohm trajectories are simply a kinematic portrayal of the evolution of the probability density. In addition, the new methods can be used to measure Schrödinger's wave function and Planck's constant. / text

Es conocido que a escalas nanométricas se debe tratar con en el problema de muchas partículas a la hora de estudiar dispositivos electrónicos. Es estos escenarios, la ecuación de Schrödinger dependiente del tiempo para muchas partículas solo se puede resolver para unos pocos grados de libertad. En este sentido, diferentes formalismos han sido desarrollados en la literatura (tales como time-dependent Density Functional Theory, Green's functions técnicas o Quantum Monte Carlo técnicas) para tratar sistemas cuánticos de muchos electrones. Estas aproximaciones modelizan de forma razonable el transporte electrónico en sistemas de muchas partículas. Una propuesta alternativa ha sido desarrollada por el Dr. Oriols para descomponer la ecuación de Schrödinger de N-partículas en un sistema de N-ecuaciones de Schrödinger para una sola partícula usando trayectorias (cuánticas) de Bohm. Basado en esta propuesta se presenta un 3D, general, versátil y dependiente del tiempo simulador de transporte de dispositivos electrónicos llamado BITLLES (Bohmian Interacting Transport for non-equiLibrium eLEctronic Structures). Las novedades que aporta el simulador BITLLES se basan en dos puntos. El primero, éste representa un modelo de transporte cuántico de electrones para muchas partículas en el cual se tiene en cuenta de forma explicita las correlaciones de Coulomb y de intercambio entre electrones usando trayectorias de Bohm. En segundo lugar, el simulador proporciona una completa información de los momentos de la corriente (i.e., DC, AC, fluctuaciones o incluso momentos mayores). A continuación resumimos las contribuciones que esta tesis aporta al desarrollo del simulador BITLLES. De esta forma, introducimos de forma explicita la interacción de intercambio entre electrones. En este contexto, mostramos como la interacción de intercambio es la responsable final para determinar la corriente total a través del sistema. Además presentamos una nueva aproximación para estudiar sistemas de muchas partículas donde los espines de los electrones tienen diferente orientación. Hasta donde llega nuestro conocimiento, es la primera vez que la interacción de intercambio es introducida de forma práctica en un simulador de transporte de electrones. Además presentamos la computación de la corriente total dependiente del tiempo en un contexto de alta frecuencia donde se tienen que tener en cuenta las variaciones del campo eléctrico dependientes del tiempo (i.e., la corriente de desplazamiento) para asegurar la conservación de la corriente. También discutimos el cálculo de la corriente total (conducción más desplazamiento) usando los teoremas de Ramo-Shockley-Pellegrini. Diferentes capacidades del simulador BITLLES como AC y fluctuaciones de la corriente se presentan para el diodo túnel resonante. También hemos usado el simulador BITLLES para testear un nuevo tipo de dispositivo nanoeléctronico diseñado para procesar señales dentro del espectro de los THz. Hemos llamado a este dispositivo Driven Tunneling Device. Se trata de un dispositivo de tres terminales donde la conductancia entre el drain y el source se controla por el terminal del gate el cual oscila a frecuencias de THz. También presentamos ejemplos prácticos de la funcionalidad de este dispositivo como un rectificador y un multiplicador de frecuencia. Finalmente, hemos desarrollado una aproximación numérica para resolver la ecuación de Schrödinger usando el modelo de tight-binding con el propósito de mejorar la descripción de la estructura de bandas del simulador BITLLES. / It is known that at nanoscale regime we must deal with the many-particle problem in order to study electronic devices. In this scenario, the time-dependent many-particle Schrödinger equation is only directly solvable for very few degrees of freedom. However, there are many electrons (degrees of freedom) in any electron device. In this sense, many-particle quantum electron formalisms (such as time-dependent Density Functional Theory, Green's functions techniques or Quantum Monte Carlo techniques) have been developed in the literature to provide reasonable approximations to model many-particle electron transport. An alternative proposal has been developed by Dr. Oriols to decompose the N-particle Schrödinger equation into a N-single particle Schrödinger equation using Bohmian trajectories. Based on this proposal a general, versatile and time-dependent 3D electron transport simulator for nanoelectronic devices, named BITLLES (Bohmian Interacting Transport for non-equiLibrium eLEctronic Structures) is presented. The novelty of the BITLLES simulator is based on two points. First, it presents a many-particle quantum electron transport model taking into account explicitly the Coulomb and exchange correlations among electrons using Bohmian trajectories. Second, it provides full information of the all current distribution moments (i.e. DC, AC, fluctuations and even higher moments). We summarize the important contributions of this thesis to the development of BITLLES simulator. Thus, we introduce explicitly the exchange correlations among electrons. In this context, we show how exchange interaction is the final responsible for determining the total current across the system. We also present a new approximation to study many-particle systems with spin of different orientations. Some practical examples are studied taking into account the exchange interaction. To the best of our knowledge, it is the first time that the exchange interaction is introduced explicitly (imposing the exchange symmetry properties directly into the many-particle wavefunction) in practical electron transport simulators. We present the computation of the time-dependent total current in the high-frequency regime where one has to compute time-dependent variations of the electric field (i.e. the displacement current) to assure current conservation. We discuss the computation of the total (conduction plus displacement) current using Bohmian trajectories and the Ramo-Shockley-Pellegrini theorems. Different capabilities of BITLLES simulator such as AC and current fluctuations are presented for Resonant Tunneling Devices. We have used the BITLLES simulator to test a new type of nanoelectronic device designed to process signals at THz regime named Driven Tunneling Device. It is a three terminal device where the drain-source conductance is controlled by a gate terminal that can oscillate at THz frequencies. We also present practical examples on the functionality of this device such as rectifier and frequency multiplier. Finally, we have developed a numerical approximation to solve the Schrödinger equation using tight-binding model to improve the band structure description of the BITLLES simulator.

We introduce the de Broglie-Bohm causal interpreation of quantum mechanics and compare it to the standard interpretation of quantum mechanics, the Copenhagen interpretation. We examine the possibility of experimentally distinguishing between the two theories, as well as the potential for the causal interpretation to more easily bridge the gap between the physics of the quantum and classical worlds. We then use the causal interpretation to construct a deterministic model of the helium atom in which the two electrons move along trajectories through space and time about a stationary nucleus. The dynamics are governed by the non-relativistic Schrödinger equation and the spin vectors of both electrons are assumed to be constant along their respective trajectories. We examine the Bohmian trajectories associated with (approximations to) eigenstates of the helium Hamiltonian as well as the trajectories associated with some non-eigenstates. We also compute an approximation to the ground state energy of the helium atom using a representation of the helium wavefunction in terms of hydrogenic eigenfunctions which is motivated by a perturbation approach.

We introduce the de Broglie-Bohm causal interpreation of quantum mechanics and compare it to the standard interpretation of quantum mechanics, the Copenhagen interpretation. We examine the possibility of experimentally distinguishing between the two theories, as well as the potential for the causal interpretation to more easily bridge the gap between the physics of the quantum and classical worlds. We then use the causal interpretation to construct a deterministic model of the helium atom in which the two electrons move along trajectories through space and time about a stationary nucleus. The dynamics are governed by the non-relativistic Schrödinger equation and the spin vectors of both electrons are assumed to be constant along their respective trajectories. We examine the Bohmian trajectories associated with (approximations to) eigenstates of the helium Hamiltonian as well as the trajectories associated with some non-eigenstates. We also compute an approximation to the ground state energy of the helium atom using a representation of the helium wavefunction in terms of hydrogenic eigenfunctions which is motivated by a perturbation approach.

The aim of this thesis was to simulate a part of a proposed experiment by Lev Vaidman by using Bohmian mechanics. To do this a numerical method for solving the Schrödinger equation and theguiding equation was created, with several ways of making the simulation more efficient.To make the simulation work more efficiently the Schrödinger equation was applied to only a small region of the whole setup. This region followed the wavefunction of significant values and could change size during the simulation. A beam splitter was constructed in the form of a thin potential barrier. The beam splitter was tested to verify that the reflected and transmitted angles agreed with expectations. A virtual detector was constructed and used for the calibration of the beam splitter to determine which potential resulted in dividing the wave packet into two wave packets of equal intensity. A fixed angle mirror was used for testing the reflection of a wave packet for the reflected angle and concluded that it agreed with the expectations for it. Testing a time dependent mirror for different frequencies and amplitudes was performed, with the result that the numerical method could be used to determine the particles’ trajectories. These results were used to construct a larger setup that was a small part of Vaidman’s proposed experiment. These setups were done in several version. All setups had one wave packet that went through one beam splitter and separated into two wave packets. These two wave packets reflected at two mirrors with different frequencies and then interfered with each other at either free space or at another beam splitter. The result of the simulation of these setups was that the particles’ trajectories could be calculated with the guiding equation. / Syftet med denna avhandling var att simulera en del av det föreslagna experimentet av Lev Vaidman med hjälp av Bohmsk mekanik. För att göra detta skapades en numerisk metod för att lösa Schrödingerekvationen och den ledande ekvationen, ”the guiding equation”, med flera sätt att effektivisera simuleringen. För att effektivisera simuleringen tillämpades Schrödingerekvationen på endast en liten region i hela uppställningen. Denna region följde vågfunktionen med betydande värden och kunde ändra storlek under simuleringen.En stråldelare konstruerades i form av en tunn potentialbarriär. Stråldelaren testades för att verifiera attde reflekterade och överförda vinklarna överensstämde med förväntningarna. En virtuell detektorkonstruerades och användes för kalibrering av stråldelaren för att bestämma vilken potential som resulterade i att vågpaketet delades in i två vågpaket med samma intensitet.En spegel med fast vinkel användes för att testa reflektionen av ett vågpaket för den reflekterade vinkeln och kom fram till att den överensstämde med förväntningarna för den. Att testa en tidsberoendespegel för olika frekvenser och amplituder utfördes med resultatet att den numeriska metoden kunde användas för att bestämma partiklarnas banor. Dessa resultat användes för att konstruera en större uppställning av ett experiment som var en liten delav Vaidmans föreslagna experiment. Dessa uppställningar gjordes i flera versioner. Alla uppställningar hade ett vågpaket som gick igenom en stråldelare och separerades i två vågpaket. Dessa två vågpaket reflekterades vid två speglar med olika frekvenser och interfererade sedan varandra antingen i en tom rymd eller vid en annan stråldelare. Resultatet av simuleringen av dessa inställningar var att partiklarnas banor kunde beräknas med ledande ekvation.

<p> Bohmian mechanics is an alternative formulation of quantum mechanics that incorporates the familiar and intuitive picture of particles moving along trajectories and yet predicts the same results as the more widely accepted Copenhagen interpretation.</p> <p> In recent years there has been renewed interest in this Bohmian view, in part for the novel approach that it suggests to certain problems, such as decay processes, both from a theoretical and computational stand point. In this thesis we focus on using the concepts introduced by the Bohmian framework as a practical computational tool.</p> <p> I evaluate a number of implementations of the Bohmian method, get a sense of their strengths and weaknesses and attempt to overcome some stability issues that arise. For problems in one-dimension (lD), accurate solutions of the time-dependent Schrodinger equation produce a wave function from which Bohmian trajectories can be computed by integrating along flux lines. For direct integration of the quantum Hamilton-Jacobi equations, the main problems that arise are related to evauating the quantum potential (QP), especially in regions of low probability density. Sufficient accuracy is required to avoid unphysical trajectory crossings. A number of interpolation schemes were investigated, and smoothed splines with special treatment of edge effects gave the best results.</p> <p> For problems in 2D the alternating direction implicit (ADI) method was employed to produce the wave function. Ways of dealing with unphysical reflections from the boundaries of a finite size domain were studied.</p> <p> The use of cellular automata, especially the lattice-Boltzmann method (LBM) were also considered. Here Bohm trajectories would be propagated by following a small set of rules. The main problem identified is that, unless a scheme can be found in which the quantum potential is self-generating from an equation of continuity, the overhead of computing the QP at each time step, is prohibitive.</p> / Thesis / Doctor of Philosophy (PhD)

Parsimony, sometime referred to as simplicity, is an effective criterion of theory choice in the case of Quantum Mechanics. The Copenhagen and Bohmian interpretations are rival theories, with the Bohmian interpretation being more parsimonious. More parsimonious theories have a higher probability of being true than less parsimonious rivals. The Bohmian interpretation should thus be preferred on these grounds.

The Socratic Method has been an educational tool ever since Socrates himself turned the marketplace of Athens into a classroom, enticing his interlocutors into dialogue whereby they could have their assumptions questioned and learn to journey towards new conceptions of knowledge and understanding. This concept has been reflected recently in a current proposal by UNESCO for educators and philosophers to find ways in which philosophy and philosophical inquiry may be approached in current education practices to enhance democratic ways of life. I draw on the UNESCO idea of philosophy as a ‘school of freedom’ and contend that not only is dialogical inquiry useful to teaching and learning, but that it is necessary. Inquiry is viewed in this way as necessarily dialogical and I draw on both Charles Peirce’s and John Dewey’s views on inquiry as being situated within the community if it is to satisfy some of the aims of the UNESCO report. This dissertation proposes a framework for Socratic pedagogy, a collaborative inquiry-based approach to teaching and learning suitable not only for formal educational settings such as the school classroom but for all educational settings. The term is intended to capture a variety of philosophical approaches to classroom practice that could broadly be described as Socratic in form. I explore three models that, I argue, make a significant contribution to Socratic pedagogy: Matthew Lipman’s Community of Inquiry, Leonard Nelson’s Socratic Dialogue, and David Bohm’s Dialogue. I also draw on the metaphors used by each of the proponents because they give an additional insight into the theoretical underpinnings of their models of dialogue for the development of Socratic thinking. Socratic pedagogy is multi-dimensional, which I argue is underpinned by generative, evaluative, and connective thinking. These terms are better placed to describe Socratic pedagogy than creative, critical, and caring thinking, because they are defined by the function they perform. It is hoped that this dissertation offers some way to show how philosophy as inquiry can contribute to educational theory and practice, while also demonstrating how it can be an effective way to approach teaching and learning. This, I contend is foundational to Socratic pedagogy.