On the classical limit of quantum mechanics (original) (raw)

A Top-down View of the Classical Limit of Quantum Mechanics

2016

Since its birth in the early twentieth century, quantum mechanics raised a number of questions and problems, many of which are still a source of lively debate. The attempts to address those issues have led to a multiplicity of interpretations and theoretical developments which have enriched the scientific knowledge about the theory. Perhaps the problem most discussed in this context is the socalled quantum measurement problem, based on the theoretical difficulty to explain how measuring devices with classical pointers are able to produce results when acting on quantum systems (von Neumann 1932, Ballentine 1990, Bub 1997). Another question that has been the subject of intensive research is the problem of the classical limit of quantum mechanics (Bohm 1951, Schlosshauer 2007). According the correspondence principle (Bohr 1920; for a recent discussion, see Bokulich 2014), there should be a limiting procedure that accounts for the classical behavior of a system in terms of the laws of quantum mechanics. The problem of classical limit consists in explaining how the classical realm "emerges" from the quantum domain. The two problems just mentioned have something in common: both point to the need for finding a link between the classical and the quantum world. Along the history of quantum mechanics, the classical limit has been approached from many different perspectives, such as those given by the Ehrenfest theorem (Ehrenfest 1927), the Wigner transform (Wigner 1932) and the deformation theory (Bayen et al. 1977, 1978). Traditionally, the problem was conceived as a matter of intertheory relation: classical mechanics should be obtained from quantum mechanics by means of the application of a mathematical limit, in a way analogous to the way in which the classical equations of motion are obtained from special relativity. However, this approach has been weakening over the past decades: at present it is recognized that the classical limit also involves some kind of physical process, which transforms the quantum states in such a way that they finally can be interpreted as classical states. This process is now known as quantum decoherence. One the main features of quantum mechanics is the superposition principle, which leads to the phenomenon of quantum interference, without classical analogue. Decoherence is viewed as a process that cancels interference and selects the candidates to classical states. The cancellation of

Quantum Mechanics As A Limiting Case of Classical Mechanics

In spite of its popularity, it has not been possible to vindicate the conventional wisdom that classical mechanics is a limiting case of quantum mechanics. The purpose of the present paper is to offer an alternative point of view in which quantum mechanics emerges as a limiting case of classical mechanics in which the classical system is decoupled from its environment.

A Continuous Transition Between Quantum and Classical Mechanics. II

Foundations of Physics, 2002

In spite of its popularity, it has not been possible to vindicate the conventional wisdom that classical mechanics is a limiting case of quantum mechanics. The purpose of the present paper is to offer an alternative formulation of classical mechanics which provides a continuous transition to quantum mechanics via environment-induced decoherence.

A Critique of the Classical Limit Problem of Quantum Mechanics

Foundations of Physics Letters, 2006

The complex quantum-classical relationship is reviewed and the inadequacy of quantum mechanical wavefunction description for the centre of mass motion of a macroscopic system is discussed. The emergence of manifest classical reality in this case is analyzed and we interpret the unphysical infinitely rapid oscillations of the wavefunction near the classical regime observed in our earlier studies as the breakdown of wavefunction description for normal macroscopic mass domain above 10 −15 g. It is contended that, production of quantum interference with large macromolecules like viruses (m ∼ 10 −12 g), as proposed by some authors, is impossible. Another testable prediction asserts that particle tracks, whenever observed, will obey classical law and offers an interesting experimental verification.

A Decoherence-Based Approach to the Classical Limit in Bohm's Theory

Foundations of Physics 53: 41 , 2023

The paper explains why the de Broglie-Bohm theory reduces to Newtonian mechanics in the macroscopic classical limit. The quantum-to-classical transition is based on three steps: (i) interaction with the environment produces effectively factorized states, leading to the formation of effective wave functions and hence decoherence; (ii) the effective wave functions selected by the environment–the pointer states of decoherence theory–will be well-localized wave packets, typically Gaussian states; (iii) the quantum potential of a Gaussian state becomes negligible under standard classicality conditions; therefore, the effective wave function will move according to Newtonian mechanics in the correct classical limit. As a result, a Bohmian system in interaction with the environment will be described by an effective Gaussian state and–when the system is macroscopic–it will move according to Newtonian mechanics.

Bohmian Mechanics as the Foundation of Quantum Mechanics

Boston Studies in the Philosophy of Science, 1996

In order to arrive at Bohmian mechanics from standard nonrelativistic quantum mechanics one need do almost nothing! One need only complete the usual quantum description in what is really the most obvious way: by simply including the positions of the particles of a quantum system as part of the state description of that system, allowing these positions to evolve in the most natural way. The entire quantum formalism, including the uncertainty principle and quantum randomness, emerges from an analysis of this evolution. This can be expressed succinctly|though in fact not succinctly enough|by declaring that the essential innovation of Bohmian mechanics is the insight that particles move! 1 Bohmian Mechanics is Minimal Is it not clear from the smallness of the scintillation on the screen that we have to do with a particle? And is it not clear, from the di raction and interference 1

A Continuous Transition Between Quantum and Classical Mechanics (I)

2001

In spite of its popularity, it has not been possible to vindicate the conventional wisdom that classical mechanics is a limiting case of quantum mechanics. The purpose of the present paper is to offer an alternative formulation of classical mechanics which provides a continuous transition to quantum mechanics via environment-induced decoherence.

Bohmian Trajectories as the Foundation of Quantum Mechanics

2009

Bohmian trajectories have been used for various purposes, including the numerical simulation of the time-dependent Schrödinger equation and the visualization of time-dependent wave functions. We review the purpose they were invented for: to serve as the foundation of quantum mechanics, i.e., to explain quantum mechanics in terms of a theory that is free of paradoxes and allows an understanding that is as clear as that of classical mechanics. Indeed, they succeed in serving that purpose in the context of a theory known as Bohmian mechanics, to which this article is an introduction.

Review Essay: Bohmian Mechanics and the Quantum Revolution

Synthese, 1996

When I was young I was fascinated by the quantum revolution: the transi-tion from classical definiteness and determinism to quantum indeterminacy and uncertainty, from classical laws that are indifferent, if not hostile, to the human presence, to quantum laws that fundamentally ...

The Quantum-like Face of Classical Mechanics

arXiv (Cornell University), 2018

It is first shown that when the Schrödinger equation for a wave function is written in the polar form, complete information about the system's quantum-ness is separated out in a single term Q, the so called 'quantum potential'. An operator method for classical mechanics described by a 'classical Schrödinger equation' is then presented, and its similarities and differences with quantum mechanics are pointed out. It is shown how this operator method goes beyond standard classical mechanics in predicting coherent superpositions of classical states but no interference patterns, challenging deeply held notions of classical-ness, quantum-ness and macro realism. It is also shown that measurement of a quantum system with a classical measuring apparatus described by the operator method does not have the measurement problem that is unavoidable when the measuring apparatus is quantum mechanical. The type of decoherence that occurs in such a measurement is contrasted with the conventional decoherence mechanism. The method also provides a more convenient basis to delve deeper into the area of quantum-classical correspondence and information processing than exists at present.