Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

BOHMIAN MECHANICS AND QUANTUM EQUILIBRIUM

Sheldon Goldstein

Published Jan 1, 1995• 2 citations
OpenAlex

Abstract

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.

Related papers

Quantum Equilibrium and the Role of Operators as Observables in Quantum Theory

Detlef Dürr, Sheldon Goldstein, Nino Zanghı̀

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.

•Jan 1, 2003•114 citations

QUANTUM EQUILIBRIUM AND THE ORIGIN OF ABSOLUTE UNCERTAINTY

Detlef D�rr, Sheldon Goldstein, Nino Zangh�

Dedicated to the memory of J. S. Bell. Abstract. The quantum formalism is a \\measurement " formalism|a phenomenological formalism describing certain macroscopic regularities. We argue that it can be regarded, and best be understood, as arising from Bohmian mechanics, which is what emerges from Schrodinger's equation for a system of particles when we merely insist that \\particles " means particles. While distinctly non-Newtonian, Bohmian mechanics is a fully deterministic theory of particles in motion, a motion choreographed by the wave function. We nd that a Bohmian universe, though deterministic, evolves in such a manner that an appearance of randomness emerges, precisely as described by the quantum formalism and given, for example, by \\ = j j 2. " A crucial ingredient in our analysis of the origin of this randomness is the notion of the eective wave function of a subsystem, a notion of interest in its own right and of relevance to any discussion of quantum theory. When the quantum formalism is regarded as arising in this way, the paradoxes and perplexities so often associated with (nonrelativistic) quantum theory simply evaporate. KEY WORDS: Quantum randomness; quantum uncertainty; hidden variables; eective wave function; collapse of the wave function; the measurement problem; Bohm's causal inter-pretation of quantum theory; pilot wave; foundations of quantum mechanics. 1.

•Nov 1, 2015•555 citations

Basic elements of Bohmian mechanics

Emmanuel E. Haven, Andrei Yu. Khrennikov

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.

Cambridge University Press eBooks•Jan 17, 2013•0 citations

A Survey of Bohmian Mechanics

Karin Berndl, M. Daumer, Sheldon Goldstein +4 more

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.

•Jan 1, 1995•68 citations

Bohmian Mechanics

Roderich Tumulka

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.

arXiv:1704.08017•Apr 26, 2017•5 citations

Bohmian Mechanics is Not Deterministic

Klaas Landsman

Abstract I argue that Bohmian mechanics (or any similar pilot-wave theory) cannot reasonably be claimed to be a deterministic theory. If one assumes the “quantum equilibrium distribution” provided by the wave function of the universe, Bohmian mechanics requires an external random oracle in order to describe the (Kolmogorov–Levin–Chaitin) algorithmic randomness properties of typical outcome sequences of long runs of repeated identical experiments (which provably follow from the Born rule). This oracle lies beyond the scope of Bohmian mechanics (or any deterministic extension thereof), including the impossibility of explaining the randomness property in question from “random” initial conditions. Thus the advantages of Bohmian mechanics over other interpretations of quantum mechanics, if any, must lie at an ontological level, and in its potential to derive the quantum equilibrium distribution and hence the Born rule.

arXiv:2202.12279•Foundations of Physics•Jul 7, 2022•1 citation
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 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.
Dedicated to the memory of J. S. Bell. Abstract. The quantum formalism is a \\measurement " formalism|a phenomenological formalism describing certain macroscopic regularities. We argue that it can be regarded, and best be understood, as arising from Bohmian mechanics, which is what emerges from Schrodinger's equation for a system of particles when we merely insist that \\particles " means particles. While distinctly non-Newtonian, Bohmian mechanics is a fully deterministic theory of particles in motion, a motion choreographed by the wave function. We nd that a Bohmian universe, though deterministic, evolves in such a manner that an appearance of randomness emerges, precisely as described by the quantum formalism and given, for example, by \\ = j j 2. " A crucial ingredient in our analysis of the origin of this randomness is the notion of the eective wave function of a subsystem, a notion of interest in its own right and of relevance to any discussion of quantum theory. When the quantum formalism is regarded as arising in this way, the paradoxes and perplexities so often associated with (nonrelativistic) quantum theory simply evaporate. KEY WORDS: Quantum randomness; quantum uncertainty; hidden variables; eective wave function; collapse of the wave function; the measurement problem; Bohm's causal inter-pretation of quantum theory; pilot wave; foundations of quantum mechanics. 1.
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.
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, 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.
Abstract I argue that Bohmian mechanics (or any similar pilot-wave theory) cannot reasonably be claimed to be a deterministic theory. If one assumes the “quantum equilibrium distribution” provided by the wave function of the universe, Bohmian mechanics requires an external random oracle in order to describe the (Kolmogorov–Levin–Chaitin) algorithmic randomness properties of typical outcome sequences of long runs of repeated identical experiments (which provably follow from the Born rule). This oracle lies beyond the scope of Bohmian mechanics (or any deterministic extension thereof), including the impossibility of explaining the randomness property in question from “random” initial conditions. Thus the advantages of Bohmian mechanics over other interpretations of quantum mechanics, if any, must lie at an ontological level, and in its potential to derive the quantum equilibrium distribution and hence the Born rule.