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Observables, measurements and phase operators from a Bohmian perspective

Martin Däumer, Sheldon Goldstein

Published Jan 1, 1993• 0 citations
OpenAlex

Abstract

Bohmian mechanics is a deterministic theory of point particles in motion. While avoiding all the paradoxes of nonrelativistic quantum mechanics, it yields the quantum formalism itself---especially the role of self-adjoint operators---as a macroscopic measurement formalism. As an "application" it is shown that much of the confusion connected with the phase operator for the electromagnetic field arises from a misunderstanding of the role of operators in quantum theory. 1 Introduction We would like to apologize for the bad title: we will try to explain why the casual use of the words "observables" and "measurements," which are on John Bell's list of bad words in his article "Against Measurement"[1], "measurement" being the worst of all, leads to much unnecessary confusion concerning the meaning of the quantum formalism. But first we introduce an even worse word: following Bell we will use the abbreviation "FAPP" for "for all practical purposes." Quantum mechanics suffers from its irreduc...

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arXiv:1206.1084•Jun 5, 2012•1 citation
Bohmian mechanics is a deterministic theory of point particles in motion. While avoiding all the paradoxes of nonrelativistic quantum mechanics, it yields the quantum formalism itself---especially the role of self-adjoint operators---as a macroscopic measurement formalism. As an "application" it is shown that much of the confusion connected with the phase operator for the electromagnetic field arises from a misunderstanding of the role of operators in quantum theory. 1 Introduction We would like to apologize for the bad title: we will try to explain why the casual use of the words "observables" and "measurements," which are on John Bell's list of bad words in his article "Against Measurement"[1], "measurement" being the worst of all, leads to much unnecessary confusion concerning the meaning of the quantum formalism. But first we introduce an even worse word: following Bell we will use the abbreviation "FAPP" for "for all practical purposes." Quantum mechanics suffers from its irreduc...
A source of much difficulty and confusion in the interpretation of quantum mechanics is a ``naive realism about operators.'' By this we refer to various ways of taking too seriously the notion of operator-as-observable, and in particular to the all too casual talk about ``measuring operators'' that occurs when the subject is quantum mechanics. Without a specification of what should be meant by ``measuring'' a quantum observable, such an expression can have no clear meaning. A definite specification is provided by Bohmian mechanics, a theory that emerges from Schrodinger's equation for a system of particles when we merely insist that ``particles'' means particles. Bohmian mechanics clarifies the status and the role of operators as observables in quantum mechanics by providing the operational details absent from standard quantum mechanics. It thereby allows us to readily dismiss all the radical claims traditionally enveloping the transition from the classical to the quantum realm---for example, that we must abandon classical logic or classical probability. The moral is rather simple: Beware naive realism, especially about operators!
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.
This note presents a concise and non-polemical comparison of several major interpretations of quantum mechanics, with a particular emphasis on the distinction between FAPP-solutions ("For All Practical Purposes'') versus ontological solutions to the measurement problem. Building on this distinction, we argue that the Contexts-Systems-Modalities (CSM) framework, supplemented by the operator-algebraic description of macroscopic contexts, provides a conceptually complete, non-FAPP ontology that naturally incorporates irreversibility and the physical structure of measurement devices. This approach differs significantly from other ontological interpretations such as Bohmian mechanics, spontaneous collapse, or many-worlds, and highlights the major role of contextual quantization in shaping quantum theory.
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.
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.