Short introduction to Bohmian mechanics Sheldon Goldstein in a Foundations of Physics paper [1] indicates (p. 341): “Since macroscopic objects are normally regarded as built out of microscopic constituents …there can be no problem of macroscopic reality per se in Bohmian mechanics.” Goldstein in the same paper remarks (p. 342) that this | ψ | 2 has “a status very much the same as that of a thermodynamic equilibrium.” Basil Hiley the closest collaborator of David Bohm, says the following (Hiley [2] (p. 2)): “What Bohm (1952a; 1952b) did was to show how to retain a description of all the usual properties of a classical world and yet remain completely within the quantum formalism.” The Bohm (1952) [3] [4] references in that quote refer to the original work of Bohm in which he sets out the basics of the Bohmian mechanics. As we will see later in the next section of this chapter, the appearance of an “additional term” in the Hamilton-Jacobi equation, is the hallmark of Bohmian mechanics and it is very often termed the “quantum potential.” Hiley [2] (p. 2) remarks that since the Bohmian momentum is a well-defined function of position and time, an ensemble of trajectories can be found when the quantum potential is non-zero. A unique, classical path is found when the quantum potential is zero. From the outset, we hope the reader can savor the beauty of this simple but very powerful result. Bohmian mechanics indeed shows this very gentle transition from quantum mechanics to classical mechanics via the value of the quantum potential.
Short introduction to Bohmian mechanics Sheldon Goldstein in a Foundations of Physics paper [1] indicates (p. 341): “Since macroscopic objects are normally regarded as built out of microscopic constituents …there can be no problem of macroscopic reality per se in Bohmian mechanics.” Goldstein in the same paper remarks (p. 342) that this | ψ | 2 has “a status very much the same as that of a thermodynamic equilibrium.” Basil Hiley the closest collaborator of David Bohm, says the following (Hiley [2] (p. 2)): “What Bohm (1952a; 1952b) did was to show how to retain a description of all the usual properties of a classical world and yet remain completely within the quantum formalism.” The Bohm (1952) [3] [4] references in that quote refer to the original work of Bohm in which he sets out the basics of the Bohmian mechanics. As we will see later in the next section of this chapter, the appearance of an “additional term” in the Hamilton-Jacobi equation, is the hallmark of Bohmian mechanics and it is very often termed the “quantum potential.” Hiley [2] (p. 2) remarks that since the Bohmian momentum is a well-defined function of position and time, an ensemble of trajectories can be found when the quantum potential is non-zero. A unique, classical path is found when the quantum potential is zero. From the outset, we hope the reader can savor the beauty of this simple but very powerful result. Bohmian mechanics indeed shows this very gentle transition from quantum mechanics to classical mechanics via the value of the quantum potential.
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 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.
Bohmian mechanics as formulated originally in 1952, has been useful in the implementation of numerical methods applied to quantum mechanics. The scientific community though has had ever since a critical thought about it. Therefore, there are still points to be clarified and rectified. The two main problems are basically: Bohmian mechanics gives a privilege role to the position representation. Secondly, the current interpretation of Bohmian trajectories has been recently proven wrong. In this context, in Chapter 2, new complex Bohmian quantities are defined; so that they allow the capacity to formulate Bohmian mechanics in any arbitrary continuous representation, for instance, the momentum representation. This Chapter is fully based on two articles, regarding the proposed complex Bohmian formulation and its extension into momentum space. Chapter 3 deals with a redefinition and reinterpretation of the Bohmian trajectories from the handling of the continuity equation, this is done without any need of additional postulates or interpretations. Also, it is proved that Bohmian mechanics is actually more than a projective aspect of the Wigner function. As a third point, Chapter 4 presents a sytematic treatment of the hydrodynamic scheme of Bohmian mechanics. Then, a brief summary of the transport equations in Bohmian mechanics is done. Next, a unified hydrodynamic treatment is found for the Bohmian mechanics. This treatment is useful to sketch, a Bohmian treatment to efficiently find the steady value of the transmission integral. In Chapter 5 conclusions of this thesis are drawn.
BOHMIAN MECHANICS AND QUANTUM EQUILIBRIUMDetlef Durr*, Sheldon Goldstein and Nino Zangh¨ ´i**Department of MathematicsRutgers UniversityNew Brunswick, NJ 089031. IntroductionWe report here on our work on Bohmian mechanics. A much more detailedpresentation will be published elsewhere [7].According to the quantum formalism, measurements performed on a quantumsystem with definite wave function typically yield random results. However, thequantum dynamics governing the evolution of the wave function over time, at leastwhen no measurement is being performed, and given, say, by Schrodinger’s equa-tion, is completely deterministic. Thus if all physical processes, including those wecall measurements, are governed by the same fundamental laws, and if one assumesthe wave function to be the complete description of the physical state, one arrivesat an inconsistency, often referred to as “the measurement problem” or the problemof the “collapse of the wave function.”Debates over this were strongly influenced by “quantum epistemology”—by theobligatory reference to the observer in a measurement, whose act of “observation”somehow creates the random outcome. In recent years, however, the growing disen-chantment with this irreducible reference to the observer has begun to assume theform of serious doubts about the validity and meaning of the quantum formalism.
Bohmian mechanics is the most naively obvious embedding imaginable of Schr\\"odinger's equation into a completely coherent physical theory. It describes a world in which particles move in a highly non-Newtonian sort of way, one which may at first appear to have little to do with the spectrum of predictions of quantum mechanics. It turns out, however, that as a consequence of the defining dynamical equations of Bohmian mechanics, when a system has wave function \\psi its configuration is typically random, with probability density \\rho given by |\\psi|^2, the quantum equilibrium distribution. It also turns out that the entire quantum formalism, operators as observables and all the rest, is a consequence of Bohmian mechanics.
Bohmian mechanics is the most naively obvious embedding imaginable of Schrödinger’s equation into a completely coherent physical theory. It describes a world in which particles move in a highly non-Newtonian sort of way, one which may at first appear to have little to do with the spectrum of predictions of quantum mechanics. It turns out, however, that as a consequence of the defining dynamical equations of Bohmian mechanics, when a system has wave function ψ its configuration is typically random, with probability density ρ given by |ψ|², the quantum equilibrium distribution. It also turns out that the entire quantum formalism, operators as observables and all the rest, naturally emerges in Bohmian mechanics from the analysis of “measurements.” This analysis reveals the status of operators as observables in the description of quantum phenomena, and facilitates a clear view of the range of applicability of the usual quantum mechanical formulas.