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Canadian Journal of Physics

The Bohm quantum potential and the classical limit of quantum mechanics

A. O. Bolivar

Published Jul 1, 2003• 6 citations
Publisher

Abstract

Using a procedure based on the limit [Formula: see text], we show that the classical limiting method based on the Bohm quantum potential (Q [Formula: see text] 0) is not necessary to characterize the classical limit of quantum mechanics. PACS No.: 03.65.Ta
DOI
10.1139/p03-063

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We discuss classical electrodynamics and the Aharonov–Bohm effect in the presence of the minimal length. In the former, we derive the classical equation of motion and the corresponding Lagrangian. In the latter, we adopt the generalized uncertainty principle (GUP) and compute the scattering cross-section up to the first-order of the GUP parameter [Formula: see text]. Even though the minimal length exists, the cross-section is invariant under the simultaneous change [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] are azimuthal angle and magnetic flux parameter. However, unlike the usual Aharonv–Bohm scattering, the cross-section exhibits discontinuous behavior at every integer [Formula: see text]. The symmetries, which the cross-section has in the absence of GUP, are shown to be explicitly broken at the level of [Formula: see text].
Conceptual and formal problems Many of the problems surrounding the interpretation of quantum mechanics stem from the absence of a sufficiently general and physically transparent analysis of the connection with the domain in which classical concepts are assumed to apply. It is the purpose of this chapter to describe with the aid of a few examples how the quantum potential approach provides a context within which the conceptual and formal issues raised by this problem may be clarified. A key question is to explain how the state or context dependence which is so characteristic of the quantum domain turns into state independence at a certain level, characterized as ‘classical’. To illustrate one aspect of the conceptual problem we recall that in Bohr's interpretation the validity of classical concepts is already presupposed since, it is suggested, it is only in terms of these that one can unambiguously communicate the results of experiments in the quantum domain. Thus, classical physics must be considered as prior to quantum mechanics and the latter is a generalization of the former in that it provides a new set of laws governing the application of classical concepts. According to this view, any procedure by which classical mechanics is recovered from quantum mechanics as a mathematical limit can only be a demonstration of consistency with the already postulated epistemological relation between the two theories and not as a ‘derivation’ of classical mechanics from quantum mechanics (George, Prigogine & Rosenfeld, 1972).
A Bose–Einstein condensation theory for any integer spin using noncommutative quantum mechanics methods is considered. The effective potential is derived as a multipolar expansion in the non-commutativity parameter [Formula: see text] and, at second order in [Formula: see text], our procedure yields to the standard dipole–dipole interaction with [Formula: see text] playing the role of the strength interaction parameter. The generalized Gross–Pitaevskii equation containing nonlocal dipolar contributions is found. For [Formula: see text] isotopes [Formula: see text] becomes [Formula: see text][Formula: see text][Formula: see text] and, thus for this value of [Formula: see text] one cannot distinguish interactions coming from non-commutativity or those of dynamical origin.