Utility: The condition, quality, or fact of being useful or beneficial (The Shorter Oxford English Dictionary) Bohmian mechanics-the causal interpretation of quantum mechanics first introduced in completed form by David Bohm in 1952 [1,2]—is generally believed to be empirically equivalent to “standard” quantum mechanics. In principle, any result obtainable in one should be obtainable in the other. It need not be true, however, that equivalent results may be obtained with equivalent ease in the two interpretations, or that they always render equally valuable physical insights.
Utility: The condition, quality, or fact of being useful or beneficial (The Shorter Oxford English Dictionary) Bohmian mechanics-the causal interpretation of quantum mechanics first introduced in completed form by David Bohm in 1952 [1,2]—is generally believed to be empirically equivalent to “standard” quantum mechanics. In principle, any result obtainable in one should be obtainable in the other. It need not be true, however, that equivalent results may be obtained with equivalent ease in the two interpretations, or that they always render equally valuable physical insights.
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.
Abstract The paper investigates David Bohm’s philosophy of science as a form of fallibilist realism that underpins his causal interpretation of quantum mechanics Bohm (Bohm 1952a, 1952b, 1984, 1985, 1988, 1989, 1990; Bohm and Hiley 1975, 1984, 1993). Bohm’s central aim is to establish the superiority of his interpretation over the standard Copenhagen framework, particularly in contrast to that of Niels Bohr. Whereas Bohr asserts that knowledge of quantum phenomena is restricted to the statistical outcomes of measurement, Bohm’s approach is distinctly realist in orientation. Responding to the challenges posed by Einstein and other physicists, Bohm relies on extra-empirical criteria such as determinism, objectivity and intelligibility. The paper develops what will be termed the methodology of intuitive intelligibility (II), consisting of four stages: (1) reformulation of the standard quantum equations, (2) physical interpretation of the mathematical symbols, (3) drawing analogies with classical laws, and (4) drawing analogies with common experience. It is argued that Bohm applies the (II) methodology consistently, with the notable exception of the ontological status of the wave function and quantum potential. In attempting to explain causal relations from a 3N-dimensional configuration space (where N is the number of particles) to familiar three-dimensional space through an appeal to ‘information’, Bohm arguably commits a category mistake. A proposed resolution—drawing on Bohm’s own holistic analogies—is supported by the recent multi-field interpretation (Romano Multi-field and Bohm’s theory. Synthese, 198(30), 10587–10609, 2021). The term fallibilist realism aptly characterizes Bohm’s stance in the philosophy of science: he repeatedly emphasizes that the causal interpretation is not intended as a final theory. Nevertheless, holism constitutes a substantive ontological commitment that he is unwilling to abandon.
This chapter provides a fully comprehensive overview of the Bohmian formulation of quantum phenomena. It deals with a historical review of the difficulties found by Louis de Broglie, David Bohm and John Bell to convince the scientific community about the validity and utility of Bohmian mechanics. The chapter presents a formal explanation of Bohmian mechanics for nonrelativistic single-particle quantum systems. It explains a didactic introduction of the Bohmian formalism and a self-contained summary for any newcomer interested in using Bohmian mechanics in their daily research activity. The quantum theory revolves around the notions of particles and waves. Einstein died in 1955, convinced that a correct reformulation of quantum theory would preserve local causality. Aerodynamics and hydrodynamics are examples of such continuous material systems. The initial formulation of quantum mechanics with Bohmian trajectories developed by de Broglie and Bohm was performed for dynamical systems with associated velocities much slower than the speed of light.
Bohmian mechanics, also known as pilot-wave theory or de Broglie-Bohm theory, is a formulation of quantum mechanics whose fundamental axioms are not about what observers will see if they perform an experiment but about what happens in reality. It is therefore called a "quantum theory without observers," alongside with collapse theories and many-worlds theories and in contrast to orthodox quantum mechanics. It follows from these axioms that in a universe governed by Bohmian mechanics, observers will see outcomes with exactly the probabilities specified by the usual rules of quantum mechanics for empirical predictions. Specifically, Bohmian mechanics asserts that electrons and other elementary particles have a definite position at every time and move according to an equation of motion that is one of the fundamental laws of the theory and involves a wave function that evolves according to the usual Schrödinger equation. Bohmian mechanics is named after David Bohm (1917-1992), who was, although not the first to consider this theory, the first to realize (in 1952) that it actually makes correct predictions.
Bohmian mechanics, also known as pilot-wave theory or de Broglie–Bohm theory, is a formulation of quantum mechanics whose fundamental axioms are not about what observers will see if they perform an experiment but about what happens in reality. It is therefore called a “quantum theory without observers,” alongside with collapse theories and many-worlds theories and in contrast to orthodox quantum mechanics. It follows from these axioms that in a universe governed by Bohmian mechanics, observers will see outcomes with exactly the probabilities specified by the usual rules of quantum mechanics for empirical predictions. Specifically, Bohmian mechanics asserts that electrons and other elementary particles have a definite position at every time and move according to an equation of motion that is one of the fundamental laws of the theory and involves a wavefunction that evolves according to the usual Schr\\"odinger equation. Bohmian mechanics is named after David Bohm (1917–1992), who was, although not the first to consider this theory, the first to realize (in 1952) that it actually makes correct predictions.
The popular impression of Bohmian mechanics is that it is standard quantum mechanics with the addition of some extra gadgets---exact particle positions and a guiding equation for particle trajectories---the advantages being that the gadgets pave the way for a resolution of the measurement problem that eschews state vector reduction while restoring the determinism lost in standard quantum mechanics. In fact, the Bohmian mechanics departs in significant ways from standard quantum mechanics. By itself this is not a basis for criticism; indeed, it makes Bohmian mechanics all the more interesting. But Bohmian mechanics is not, as the popular impression would have it, empirically equivalent to standard quantum mechanics in terms of probabilistic predictions for the outcomes of measurements of quantum observables. Indeed, in physically important applications to systems for which standard quantum mechanics delivers empirically well-confirmed probabilistic predictions, the sophisticated form of Bohmian mechanics designed to prove the global existence of Bohmian particle trajectories fails to deliver unequivocal predictions---of even a probabilistic variety---for the future behavior of said systems. Possible responses to this lacuna are discussed.