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arXiv:1801.02499

The Quantum-like Face of Classical Mechanics

Partha Ghose

Published Jan 5, 2018• 1 citation
arXivHTMLPDF

Abstract

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 {\em 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.

DOI
10.48550/arxiv.1801.02499

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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 {\em quantum-ness} is separated out in a single term QQQ, 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.
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 measurement problem in quantum mechanics originates in the inability of the Schrödinger equation to predict definite outcomes of measurements. This is due to the lack of objectivity of the eigenstates of the measuring apparatus. Such objectivity can be achieved if a unified realist conceptual framework can be formulated in terms of wave functions and operators acting on them for both the quantum and classical domains. Such a framework was proposed and an equation for the wave function (13, 14) smoothly interpolates between the quantum and classical limits. The arguments leading to the equation are clarified in this paper, and the theory is developed further. The measurement problem in quantum mechanics is then briefly reviewed and re-examined from the point of view of this theory, and it is shown how the classical limit of the wave function of the measuring apparatus leads to a natural solution of the problem of definite measurement outcomes without the need for either collapse or pragmatic thermodynamic arguments. This is consistent with Bohr's emphasis on the primacy of classical concepts and classical measuring devices. Possible tests of the theory using low-dimensional systems such as quantum dots are indicated.
The laws of quantum mechanics differ significantly from the laws of classical mechanics. The mathematical apparatus of quantum mechanics is based on the amazing Schrödinger equation. This is a specific partial differential equation with respect to a complex-valued function called the wave function. This characteristic allows, in particular, estimating the probability of finding a quantum mechanical particle in a given volume. We establish the Schrödinger equation for a free particle and for a particle moving in an external field. Then the problem of passing a potential barrier is investigated. It is shown that, in contrast to the classical case, even a relatively slow quantum mechanical particle is capable of overcoming a sufficiently high potential barrier. The Appendix considers the concept of a normalized wave function and analyzes the movement of a particle in a potential well with infinite walls.
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This book develops and simplifies the concept of quantum mechanics based on the postulates of quantum mechanics. The text discusses the technique of disentangling the exponential of a sum of operators, closed under the operation of commutation, as the product of exponentials to simplify calculations of harmonic oscillator and angular momentum. Based on its singularity structure, the Schrödinger equation for various continuous potentials is solved in terms of the hypergeometric or the confluent hypergeometric functions. The forms of the potentials for which the one-dimensional Schrödinger equation is exactly solvable are derived in detail. The problem of identifying the states of two-level systems which have no classical analogy is addressed by going beyond Bell-like inequalities and separability. The measures of quantumness of mutual information in two two-level systems is also covered in detail.
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