Classical Mechanics as Nonlinear Quantum Mechanics (original) (raw)

Nonlinear Generalisation of Quantum Mechanics

2021

It is known since Madelung that the Schrödinger equation can be thought of as governing the evolution of an incompressible fluid, but the current theory fails to mathematically express this incompressibility in terms of the wavefunction without facing problem. In this paper after showing that the current definition of quantum-mechanical momentum as a linear operator is neither the most general nor a necessary result of the de Broglie hypothesis, a new definition is proposed that can yield both a meaningful mathematical condition for the incompressibility of the Madelung fluid, and nonlinear generalisations of Schrödinger and Klein-Gordon equations. The derived equations satisfy all conditions that are expected from a proper generalisation: simplification to their linear counterparts by a well-defined dynamical condition; Galilean and Lorentz invariance (respectively); and signifying only rays in the Hilbert space.

Quantum Mechanics as a Simple Generalization of Classical Mechanics

Foundations of Physics, 2009

A motivation is given for expressing classical mechanics in terms of diagonal projection matrices and diagonal density matrices. Then quantum mechanics is seen to be a simple generalization in which one replaces the diagonal real matrices with suitable Hermitian matrices.

A Solvable Model of a Nonlinear extension of Quantum Mechanics

arXiv (Cornell University), 2022

We introduce a particular nonlinear generalization of quantum mechanics which has the property that it is exactly solvable in terms of the eigenvalues and eigenfunctions of the Hamiltonian of the usual linear quantum mechanics problem. We hope that this simple example will elucidate some of the issues of interpreting nonlinear generalization of quantum mechanics that have been put forth to resolve questions about quantum measurement theory.

The Classical Schrödinger Equation

Pioneer Journal of Mathematical Physics and its Applications, 2014

Using a simple geometrical construction based upon the linear action of the Heisenberg-Weyl group we deduce a new nonlinear Schrödinger equation that provides an exact dynamic and energetic model of any classical system whatsoever, be it integrable, nonintegrable or chaotic. Within our model classical phase space points are represented by equivalence classes of wavefunctions that have identical position and momentum expectation values. Transport of these equivalence classes is effected in a manner that avoids dispersion and thereby leads to a system of wavefunction dynamics such that the expectation values track classical trajectories precisely for arbitrarily long times. Interestingly, the value of proves immaterial for the purpose of constructing this alternative version of classical mechanics. The new feature which does mediate concerns a surprising embedding of Berry's phase within ordinary classical mechanics. Some interesting problems are exposed concerning inclusion of the projection postulate within this model nonlinear system and we discover a remarkable route for the recovery of the ordinary linear theory.

Linear quantum theory and its possible nonlinear generalizations

Annals of Physics, 1994

We show that the Schrödinger equation may be derived as a consequence of three postulates: 1) the hamiltonian formalism 2) a conformal structure 3) a projective structure. These suffice to deduce the geometrical structure of Hilbert space also. Furthermore, the quantum mechanical action principle, and unitary propagator, are reduced to special cases of classical results. Thereafter we explore the relaxation of these postulates. Of the possible generalizations only one appears physically fruitful. This is obtained from a simple gauge freedom of the standard theory. The result is a nonlinear quantum theory that has been the subject of recent interest. We consider the empirical status of this theory, and the problem of its interpretation. It is suggested that the only remaining option is to consider nonperturbative nonlinearities in the context of the quantum measurement problem. This is advanced as a principle of constraint.

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.

0 Perspectives on Nonlinearity in Quantum Mechanics

2000

Earlier H.-D. Doebner and I proposed a family of nonlinear time-evolution equations for quantum mechanics associated with certain unitary representations of the group of diffeomorphisms of physical space. Such nonlinear Schrödinger equations may describe irreversible, dissipative quantum systems. We subsequently introduced the group of nonlinear gauge transformations necessary to understand the resulting quantum theory, deriving and interpreting gauge-invariant parameters that characterize (at least partially) the physical content. Here I first review these and related results, including the coupled nonlinear Schrödinger-Maxwell theory, for which I also introduce the gauge-invariant (hy-drodynamical) equations of motion. Then I propose a further, radical generalization. An enlarged group G of nonlinear transformations, modeled on the general linear group GL(2, R), leads to a beautiful, apparently unremarked symmetry between the wave function's phase and the logarithm of its ampl...

Quantum Mechanics As A Classical Theory I: Non-relativistic Theory, quant-ph/9503020

2012

The objective of this series of three papers is to axiomatically derive quantum mechanics from classical mechanics and two other basic axioms. In this first paper, Schroendiger’s equation for the density matrix is fist obtained and from it Schroedinger’s equation for the wave functions is derived. The momentum and position operators acting upon the density matrix are defined and it is then demonstrated that they commute. Pauli’s equation for the density matrix is also obtained. A statistical potential formally identical to the quantum potential of Bohm’s hidden variable theory is introduced, and this quantum potential is reinterpreted through the formalism here proposed. It is shown that, for dispersion free ensembles, Schroedinger’s equation for the density matrix is equivalent to Newton’s equations. A general non-ambiguous procedure for the construction of operators which act upon the density matrix is presented. It is also shown how these operators can be reduced to those which a...

Classical Mechanics as a Fundamental Law of Quantum Mechanics

n our previous paper, we showed that the so-called quantum entanglement also exists in classical mechanics. The inability to measure this classical entanglement was rationalized with the definition of a classical observer which collapses all entanglement into distinguishable states. It was shown that evidence for this primary coherence is Newton’s third law. However, in reformulating a "classical entanglement theory" we assumed the existence of Newton’s second law as an operator form where a force operator was introduced through a Hilbert space of force states. In this paper, we derive all related physical quantities and laws from basic quantum principles. We not only define a force operator but also derive the classical mechanic's laws and prove the necessity of entanglement to obtain Newton’s third law.

Consistent Interpretation of Quantum and Classical Mechanics

2023

Interpreting quantum mechanics is a hard problem basically because it means explaining why and how the mathematics exploited to formulate wave-particle duality are related to observations or reality in classical physics. Consequently, interpretation of quantum mechanics and its formalism should involve proper physical mathematics, physical logic and classical physics, which is not the case from the Copenhagen interpretation. Here, we shall revisit all the postulates of quantum mechanics with proper physics and physical logic and reconstruct them to establish the less-complex interpretation of quantum mechanics with additional new postulates from classical physics. A new quantum mechanical postulate (Postulate 9) is also introduced to understand the meaning of the forbidden gap between discrete energy levels and/or electron orbitals in atoms that is valid for molecules and condensed matter based on quantum mechanics that is technically well-defined and experimentally observable. Here, we tackle all the problems arise from the Copenhagen interpretation systematically by revising and/or extending them with proper classical and quantum physics without violating established experiments and without proposing ideas that violate physical reality.