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On the Classical Limit of Quantum Mechanics  [PDF]
Valia Allori,Nino Zanghì
Physics , 2001,
Abstract: Contrary to the widespread belief, the problem of the emergence of classical mechanics from quantum mechanics is still open. In spite of many results on the $\h \to 0$ asymptotics, it is not yet clear how to explain within standard quantum mechanics the classical motion of macroscopic bodies. In this paper we shall analyze special cases of classical behavior in the framework of a precise formulation of quantum mechanics, Bohmian mechanics, which contains in its own structure the possibility of describing real objects in an observer-independent way.
Classical and Quantum Nambu Mechanics  [PDF]
Thomas Curtright,Cosmas Zachos
Physics , 2002, DOI: 10.1103/PhysRevD.68.085001
Abstract: The classical and quantum features of Nambu mechanics are analyzed and fundamental issues are resolved. The classical theory is reviewed and developed utilizing varied examples. The quantum theory is discussed in a parallel presentation, and illustrated with detailed specific cases. Quantization is carried out with standard Hilbert space methods. With the proper physical interpretation, obtained by allowing for different time scales on different invariant sectors of a theory, the resulting non-Abelian approach to quantum Nambu mechanics is shown to be fully consistent.
Quantum mechanics and classical trajectories  [PDF]
Christoph N?lle
Physics , 2010,
Abstract: The classical limit $\hbar$->0 of quantum mechanics is known to be delicate, in particular there seems to be no simple derivation of the classical Hamilton equation, starting from the Schr\"odinger equation. In this paper I elaborate on an idea of M. Reuter to represent wave functions by parallel sections of a flat vector bundle over phase space, using the connection of Fedosov's construction of deformation quantization. This generalizes the ordinary Schr\"odinger representation, and allows naturally for a description of quantum states in terms of a curve plus a wave function. Hamilton's equation arises in this context as a condition on the curve, ensuring the dynamics to split into a classical and a quantum part.
Quantum mechanics as an approximation of statistical mechanics for classical fields  [PDF]
Andrei Khrennikov
Physics , 2005, DOI: 10.1063/1.2158721
Abstract: We show that, in spite of a rather common opinion, quantum mechanics can be represented as an approximation of classical statistical mechanics. The approximation under consideration is based on the ordinary Taylor expansion of physical variables. The quantum contribution is given by the term of the second order. To escape technical difficulties, we start with the finite dimensional quantum mechanics. In our approach quantum mechanics is an approximative theory. It predicts statistical averages only with some precision. In principle, there might be found deviations of averages calculated within the quantum formalism from experimental averages (which are supposed to be equal to classical averages given by our model).
QUANTUM MECHANICS, RELATIVITY AND THE CLASSICAL THEORIES
Anamitra Palit
International Journal of Economics and Research , 2012,
Abstract: This paper seeks to review certain salient aspects of Quantum Mechanics in the light of the Classical theories.. There is also an effort to find an alliance between Quantum Mechanics and Relativity based on the Fourier Transforms. This leads to the theoretical prediction of the gravitons and the “Otherons”. The finiteness or barrier limitations of physical quantities has been discussed with the help of the Taylor series.
Ergodic hypothesis in classical statistical mechanics
Oliveira, César R. de;Werlang, Thiago;
Revista Brasileira de Ensino de Física , 2007, DOI: 10.1590/S1806-11172007000200003
Abstract: an updated discussion on physical and mathematical aspects of the ergodic hypothesis in classical equilibrium statistical mechanics is presented. then a practical attitude for the justification of the microcanonical ensemble is indicated. it is also remarked that the difficulty in proving the ergodic hypothesis should be expected.
Classical Analytical Mechanics and Entropy Production  [PDF]
J. Silverberg,A. Widom
Physics , 2007, DOI: 10.1119/1.2772279
Abstract: The usual canonical Hamiltonian or Lagrangian formalism of classical mechanics applied to macroscopic systems describes energy conserving adiabatic motion. If irreversible diabatic processes are to be included, then the law of increasing entropy must also be considered. The notion of entropy then enters into the general classical mechanical formalism. The resulting general formulation and its physical consequences are explored.
Classical and Quantum Mechanics of Anyons  [PDF]
G. Date,M. V. N. Murthy,Radhika Vathsan
Physics , 2003,
Abstract: We review aspects of classical and quantum mechanics of many anyons confined in an oscillator potential. The quantum mechanics of many anyons is complicated due to the occurrence of multivalued wavefunctions. Nevertheless there exists, for arbitrary number of anyons, a subset of exact solutions which may be interpreted as the breathing modes or equivalently collective modes of the full system. Choosing the three-anyon system as an example, we also discuss the anatomy of the so called ``missing'' states which are in fact known numerically and are set apart from the known exact states by their nonlinear dependence on the statistical parameter in the spectrum. Though classically the equations of motion remains unchanged in the presence of the statistical interaction, the system is non-integrable because the configuration space is now multiply connected. In fact we show that even though the number of constants of motion is the same as the number of degrees of freedom the system is in general not integrable via action-angle variables. This is probably the first known example of a many body pseudo-integrable system. We discuss the classification of the orbits and the symmetry reduction due to the interaction. We also sketch the application of periodic orbit theory (POT) to many anyon systems and show the presence of eigenvalues that are potentially non-linear as a function of the statistical parameter. Finally we perform the semiclassical analysis of the ground state by minimizing the Hamiltonian with fixed angular momentum and further minimization over the quantized values of the angular momentum.
The Dirac Equation in Classical Statistical Mechanics  [PDF]
G. N. Ord
Physics , 2002, DOI: 10.1063/1.1524581
Abstract: The Dirac equation, usually obtained by `quantizing' a classical stochastic model is here obtained directly within classical statistical mechanics. The special underlying space-time geometry of the random walk replaces the missing analytic continuation, making the model `self-quantizing'. This provides a new context for the Dirac equation, distinct from its usual context in relativistic quantum mechanics.
On the classical limit of the hyperbolic quantum mechanics  [PDF]
Andrei Yu. Khrennikov
Physics , 2004,
Abstract: We demonstrated that classical mechanics have, besides the well known quantum deformation, another deformation -- so called hyperbolic quantum mechanics. The classical Poisson bracket can be obtained as the limit $h\to 0$ not only of the ordinary Moyal bracket, but also hyperbolic analogue of the Moyal bracket. Thus there are two different deformations of classical phase-space: complex Hilbert space and hyperbolic Hilbert space (module over a so called hyperbolic algebra -- the two dimensional Clifford algebra). To prove the correspondence principle we use the calculus over the hyperbolic algebra similar to functional superanalysis of Vladimirov-Volovich. Ordinary (complex) and hyperbolic quantum mechanics are characterized by two types of interference perturbation of the classical formula of total probability: ordinary $\cos$-interference and hyperbolic $\cosh$-interference.
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