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American Journal of Physics

Mixed States in Classical Mechanics

N. Rosen

Published Feb 1, 1965• 11 citations
Publisher

Abstract

It is pointed out that the nonlinear equation which corresponds to the Schrödinger equation in the classical limit, and which is equivalent to the Hamilton-Jacobi equation, has pure state solutions describing particle ensembles of a certain kind. These pure states may be combined to form mixed states corresponding to ensembles of a more general sort. An example is given in the case of reflection from a potential barrier. The existence of mixtures of states, without the superpositions like those in quantum mechanics, helps one to understand how a classical system can serve as a measuring instrument.

DOI
10.1119/1.1971282

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It is pointed out that the nonlinear equation which corresponds to the Schrödinger equation in the classical limit, and which is equivalent to the Hamilton-Jacobi equation, has pure state solutions describing particle ensembles of a certain kind. These pure states may be combined to form mixed states corresponding to ensembles of a more general sort. An example is given in the case of reflection from a potential barrier. The existence of mixtures of states, without the superpositions like those in quantum mechanics, helps one to understand how a classical system can serve as a measuring instrument.
It is pointed out that, if classical mechanics is to be regarded as a limiting case of quantum mechanics, then it must admit superpositions of states, which are generally regarded as unacceptable. To avoid this it is proposed that classical mechanics be described by a nonlinear equation which is equivalent to the Hamilton-Jacobi equation and is not always the limit of the Schrödinger equation. It is conjectured that the transition from the wave-mechanical equation to the classical equation is characterized by a mass m0 = ħc/γ12 ≈ 10−5g. This nonlinear equation may help one to understand the process of measurement of a quantum-mechanical system by means of a classical measuring instrument.
Some recent experiments claim to show that any model in which a quantum state represents mere information about an underlying physical reality of the system must make predictions which contradict those of quantum theory. The present work undertakes to investigate the issue of reality, treading a more fundamental route from the Hamilton-Jacobi equation of classical mechanics to the Schrodinger equation of quantum mechanics. Motivation for this is a similar approach from the eikonal equation in geometrical optics to the wave equation in electromagnetic theory. We rewrite the classical Hamilton-Jacobi equation as a wave equation and seek to generalise de Broglie's wave particle duality by demanding that both particle and light waves have the freedom of being described by any square-integrable function. This generalisation, which allows superposition also for matter wave functions, helps us to obtain the Schrodinger equation, whose solution can be seen to be as much objective as the classical mechanics wave function. Several other equations which one writes in quantum mechanics, including the eigenvalue equations for observables, series expansion of energy states in terms of eigenstates of observables other than energy, etc., can be written in the classical case too. Absence of any collapse of the wave function, entanglement, etc. in the classical realm have their origin in the nonlinearity of the classical wave equation. These considerations indicate that many of the puzzles in quantum mechanics are present also in classical mechanics in a dormant form, which fact shall help to demystify quantum mechanics to a great extent.
The Hamilton-Jacobi equation of classical mechanics is approached as a model reduction of conservative particle mechanics where the velocity degrees-of-freedom are eliminated. This viewpoint allows an extension of the association of the Hamilton-Jacobi equation from conservative systems to general Newtonian particle systems involving non-conservative forces, including dissipative ones. A geometric optics approximation leads to a dissipative Schrödinger equation, with the expected limiting form when the associated classical force system involves conservative forces.
Any time-dependent solution of Schrödinger equation may be always correlated to a solution of Hamilton equations or to a statistical combination of their solutions; only the set of corresponding solutions is somewhat smaller (due to existence of quantization). There is not any reason to the physical interpretation according to Copenhagen alternative as Bell's inequalities are valid in the classical physics only (and not in any alternative based on Schrödinger equation). The advantage of Schrödinger equation consists then in that it enables to represent directly the time evolution of a statistical distribution of classical initial states (which is usual in collision experiments). The Schrödinger equation (without assumptions added by Bohr) may then represent the common physical theory for microscopic as well as macroscopic physical systems. However, together with the last possibility the solutions of Schrödinger equation may be helpful also in analyzing the influence of other statistically distributed properties (e.g., spin orientations or space structures) of individual matter objects forming a corresponding physical system, which goes in principle beyond the classical physics. In any case, the contemporary quantum theory represents the phenomenological approximative description of some matter characteristics only, without providing any insight into quantum mechanism emergence. In such a case it is necessary to take into account more detailed properties at least of some involved objects.