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 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.
Perhaps because of the popularity that trajectory-based methodologies have always had in Chemistry and the important role they have played, Bohmian mechanics has been increasingly accepted within this community, particularly in those areas of the theoretical chemistry based on quantum mechanics, e.g., quantum chemistry, chemical physics, or physical chemistry. From a historical perspective, this evolution is remarkably interesting, particularly when the scarce applications of Madelung's former hydrodynamical formulation, dating back to the late 1960s and the 1970s, are compared with the many different applications available at present. As also happens with classical methodologies, Bohmian trajectories are essentially used to described and analyze the evolution of chemical systems, to design and implement new computational propagation techniques, or a combination of both. In the first case, Bohmian trajectories have the advantage that they avoid invoking typical quantum-classical correspondence to interpret the corresponding phenomenon or process, while in the second case quantum-mechanical effects appear by themselves, without the necessity to include artificially quantization conditions. Rather than providing an exhaustive revision and analysis of all these applications (excellent monographs on the issue are available in the literature for the interested reader, which can be consulted in the bibliography here supplied), this Chapter has been prepared in a way that it may serve the reader to acquire a general view (or impression) on how Bohmian mechanics has permeated the different traditional levels or pathways to approach molecular systems in Chemistry: electronic structure, molecular dynamics and statistical mechanics. This is done with the aid of some illustrative examples -- theoretical developments in some cases and numerical simulations in other cases.
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.
Abstract Despite the successful development of quantum chemistry, the deep breakdown between the conceptual basis of standard quantum mechanics and some implicit ontological assumptions of quantum chemistry is only rarely acknowledged. In fact, in the framework of standard quantum mechanics, quantum systems are not individual objects in the philosophical sense: they cannot be individualized by their positions in space and time and, as a consequence, they have no trajectories. By contrast, the basic magnitude in quantum chemistry is the electronic density, which is commonly interpreted in terms of the definite positions occupied by the individual electron in its motion around the nucleus. In the present paper, it will be argued that Bohmian Mechanics offers an ontological picture that is much closer to the quantum chemists’ way of conceiving their research field.
The quantum trajectory method in the hydrodynamical formulation of Madelung-Bohm-Takabayasi quantum mechanics is an example of showing the cognitive importance of scientific illustrations and metaphors in computational quantum chemistry and electrical engineering. The method involves several numerical schemes of solving a set of hydrodynamical equations of motion for probability density fluids, based on the propagation of those probability density trajectories. The quantum trajectory method gives rise to, for example, an authentic quantum electron transport theory of motion to, among others, classically-minded applied scientists who probably have less of a commitment to traditional quantum mechanics. They were not the usual audience of quantum mechanics and simply choose to use a non-Copenhagen type interpretation to their advantage. Thus, the metaphysical issues physicists had a trouble with are not the main concern of the scientists. With the advantages of a visual and illustrative trajectory, the quantum theory of motion by Bohm effectively bridges quantum and classical physics, especially, in the mesoscale domain. Without having an abrupt shift in actions and beliefs from the classical to the quantum world, scientists and engineers are able to enjoy human cognitive capacities extended into the quantum mechanical domain.
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.
This chapter presents a brief account on both approaches, analytic and synthetic, within chemistry. It reviews the different levels at which molecular systems are traditionally studied. The chapter explores the Born–Oppenheimer approximation and then revisiting the methodologies available to deal with both electronic configuration and dynamics, distinguishing in this latter case between molecular dynamics for a few degree-of-freedom systems and statistical mechanics. It focuses on some fundamental aspects of Bohmian mechanics and describes some specific illustrations of the application of Bohmian mechanics to problems typical of the levels mentioned. Molecular systems typically include a rather large number of degrees of freedom, which makes unaffordable their direct study. In the wave model each atom is assumed to be a point-like, positively charged nucleus surrounded by an electronic cloud, which is associated with a certain orbital and represents the electron probability distribution in configuration space.