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Classical limit of quantum mechanics
... According to the developed strategy, the proposition appeared is: if the macroscopic physical systems are only the special case of a quantum-mechanical systems (e.g. like in von Neumann's theory of the measurement problem) than it is possible to observe , under specified conditions, their quantum mechanical behavior [149,150]. The behavior of a droplet-film structure submerged into the droplet homophase or double emulsion, includeing its formation–existence–destruction states, described in this paper, will be considered as a close to the representative open macroscopic quantum system (OMQS) under the specified conditions. ...
... The behavior of a droplet-film structure submerged into the droplet homophase or double emulsion, includeing its formation–existence–destruction states, described in this paper, will be considered as a close to the representative open macroscopic quantum system (OMQS) under the specified conditions. Hence, OMQS are quantum subsystems, i.e. open quantum systems that are in inevitable permanent interaction with other physical systems, which may be named environment [150]. Does the theory of electroviscoelasticity, here presented, may be useful in discussion and/or further elucidation related to the problems of the experimental and theoretical status of decoherence? ...
... Does the theory of electroviscoelasticity, here presented, may be useful in discussion and/or further elucidation related to the problems of the experimental and theoretical status of decoherence? Some needed definitions [150]: 1. The choice of an OMQS has to be in accordance to the criteria that confirm its description by the motion equation in a " classical domain " . ...
This brief extracted review presents the recent development in basic and applied science and engineering of finely dispersed particles and related systems in general, but more profound and in-depth treatise are related to the liquid-liquid finely dispersed systems, i.e. emulsions and double emulsions. Twenty-five years ago, the idea, at first very fogy, came out from the pilot plant experiments related to the extraction Of uranium from wet phosphoric acid. In particular the solution of the entrainment problems, breaking of emulsions/double emulsions, as the succession of the extraction and stripping operations/processes, was performed In this pilot plant, secondary liquid-liquid phase separation loop was designed and carried out. The loop consisted of a lamellar coalescer and four flotation cells in series. Central equipment in the loop, relevant to this investigation, was the lamellar coalescer. The phase separation in this equipment is based on the action of external forces of mechanical and/or electrical origin, while adhesive processes at the inclined filling plates occur. Since many of related processes, e.g. adhesive processes, rupture processes and coalescence, were not very well understood, deeper research of these events and phenomena was a real scientific challenge.
... The situation is additionally complicated by the existence of different schools of quantum mechanics, arguing about physical-epistemological status of the so-called collapse (reduction) of the wave function. In this respect the situation is not much better today, and it can be said freely that the problem of universal validity of quantum mechanics is still open [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. To this end, Primas [16] emphasizes the following. ...
... This is the case in most situations in quantum chemistry, with Schrödinger's equation applied to the explored closed many-atomic quantum system with appropriate boundary conditions and adopted computational approximations (giving rise to stationary ground and excited electronic-vibrational energy eigenstates of all possible many-atomic isomers, corresponding to the minimum of the electronic potential hypersurface, depicted in Figure 1, for ground electronic and corresponding excited vibrational energy eigenstates) [29,30]. It should be noted that Schrödinger's equation cannot apply to nonstationary excitations and relaxations of the many-atomic quantum system, not only in between different isomers but also within the same isomer-when quantum deexcitation/decoherence must apply to nonpotential interaction of the open manyatomic quantum system (nondescribable fully by its self-Hamiltonian) with its quantum environment (generally field-related, including vacuum) [12,13]. 4. The similar Hopfield-like quantum-holographic picture might also be applied to individual acupuncture system [41,42] (with quantum-like macroscopic resonances, fenomenologically observed in microwave resonance therapy [50,51], which implies corresponding bipartition QS acu + QE acu , i.e., that acupuncture system has macroscopic open quantum structure with dynamically coupled electrons all along macroscopic network of acupuncture channels [41,42,50]). ...
In the context of the macroscopic quantum phenomena of the second kind, we hereby seek for a solution-in-principle of the long standing problem of the polymer folding, which was considered by Levinthal as (semi)classically intractable. To illuminate it, we applied quantum-chemical and quantum decoherence approaches to conformational transitions. Our analyses imply the existence of novel macroscopic quantum biomolecular phenomena, with biomolecular chain folding in an open environment considered as a subtle interplay between energy and conformation eigenstates of this biomolecule, governed by quantum-chemical and quantum decoherence laws. On the other hand, within an open biological cell, a system of all identical (noninteracting and dynamically noncoupled) biomolecular proteins might be considered as corresponding spatial quantum ensemble of these identical biomolecular processors, providing spatially distributed quantum solution to a single corresponding biomolecular chain folding, whose density of conformational states might be represented as Hopfield-like quantum-holographic associative neural network too (providing an equivalent global quantum-informational alternative to standard molecular-biology local biochemical approach in biomolecules and cells and higher hierarchical levels of organism, as well).
... As a first example we consider an ensemble of free particles that are distributed at t = 0 according to a probability density (11) centered at r k (0) = 0 [set r k (t) = 0 in Eq. (11)]. The initial value for S(x, t) is given by S(x, 0) = p 0,k x k , i.e., S(x, t) fulfills at t = 0 the deterministic relation (20). These initial values describe for small a localized, classical particle in the sense that there is no uncertainty with respect to position or momentum. ...
... Summarizing this section, we found three potentials V (x) ∼ x n , n = 0, 1, 2, which allow for a derivation of NM from QT in the limit → 0. For these potentials, equations of motion forx k ,p k exist, as mentioned already. Home and Sengupta [20] have shown that for these potentials the form of the quantum-mechanical solution may be obtained with the help of the classical Liouville theorem. ...
An analysis is made of the relation between quantum theory and classical
mechanics, in the context of the limit ℏ-->0. Several ways in
which this limit may be performed are considered. It is shown that
Schrödinger's equation for a single particle moving in an external
potential V does not, except in special cases, lead, in this limit, to
Newton's equation of motion for the particle. This shows that classical
mechanics cannot be regarded as emerging from quantum mechanics--at
least in this sense--upon straightforward application of the limit
ℏ-->0.
... Quantum Brownian motion (QBM, Caldeira and Leggett 1983) is paradigmatic for the field of open quantum systems theory (Breuer and Petruccione 2002). Description of quantum decoherence (Giulini et al 1996, Dugić 2004 as well as modeling of -quantum dissipation‖ is directly provided for QBM as a realistic physical situation with the well-defined classical counterpart. The usefulness of the QBM model places the model at the heart of applications regarding the nano-and mesoscopic systems and some artificial setups as well as regarding the related emerging technologies, e.g. ...
We derive explicit expressions for the first and second moments as well as the correlation function for a planar (one-dimensional) quantum Brownian rotator placed in the external harmonic potential. Our results directly provide the standard deviations for the azimuthal angle and the canonically conjugate angular momentum for the rotator. We find that there are some significant physical differences between this model and the free rotator model, which is well investigated in the literature.
... Summarizing this section, we found three potentials V (x) ∼ x n , n = 0, 1, 2 which allow for a derivation of NM from QT in the limit → 0. For these potentials equations of motion for ¯ x k , ¯ p k exist, as mentioned already. Home and Sengupta [9] have shown that for these potentials the form of the quantum-mechanical solution may be obtained with the help of the classical Liouville theorem. ...
An analysis is made of the relation between quantum theory and classical
mechanics, in the context of the limit . Several ways in which
this limit may be performed are considered. It is shown that Schr\"odinger's
equation for a single particle moving in an external potential V does not,
except in special cases, lead, in this limit, to Newton's equation of motion
for the particle. This shows that classical mechanics cannot be regarded as
emerging from quantum mechanics-at least in this sense-upon straightforward
application of the limit .
... Thus, the comparison between these two mechanics is operationally unambiguous provided one compares their statistical predictions for the dynamical evolutions of the same given initial ensemble. It is in this spirit that we adopt the scheme used in this paper[16] where the quantum and the classical evolutions are compared by starting from the same initial ensemble that has the specified position and momentum distributions obtained from a given wave function. While the quantum evolution is in accordance with the Schrödinger equation, the classical evolution of the given initial ensemble is calculated in terms of the classical phase space dynamics based on Liouville's equation. ...
The classical limit problem of quantum mechanics is revisited on the basis of
a scheme that enables a quantitative study of the way the quantum-classical
agreement emerges while going through the intermediate mass range between the
microscopic and the macroscopic domains. As a specific application of such a
scheme, we investigate the classical limit of a quantum time distribution - an
area of study that has remained largely unexplored. For this purpose, we focus
on the arrival time distribution in order to examine the way the observable
results pertaining to the quantum arrival time distribution which is defined in
terms of the probability current density gradually approach the relevant
classical statistical results for an ensemble that corresponds to a Gaussian
wave packet evolving in a linear potential.
... No entanto, Ballentine et al. [6,7] (veja também [8,9]) recentemente demonstraram que as condições de validade do teorema de Ehrenfest não são nem necessárias nem suficientes para caracterizar o limite clássico da mecânica quântica. O método WKB1011121314, por sua vez, parte de uma dada solução ψ = e ı[ ...
Neste trabalho, apresentamos um procedimento alternativo para o cálculo do limite clássico ( ® 0) das equações de movimento da mecânica quântica. Comparamos nosso método com o teorema de Ehrenfest, a aproximação WKB e com o potencial quântico de Bohm Q ® 0. Segue, também, a tradução de um artigo do Einstein sobre a problemática da conexão entre as teorias clássica e quântica.
We have often found among many of our students and colleagues the common idea that the mathematical expression for a physical quantity that is essentially of quantum nature must contain a dependence on . Conversely, a phenomenon described by classical physics should contain no explicit reference to . However, the problem of a particle encountering a discontinuous potential step, which is one of the simplest examples in quantum mechanics, contradicts this common thought: even when the particle carries enough kinetic energy to go across the step, the resulting expression for the reflection probability is non-zero—a purely quantum phenomenon—and yet it contains no reference to . We show that the absence of in this purely quantum expression is due to the idealised limit in which the potential rises sharply at a single position, thus losing any reference to a length dimension in the problem. To address the correct classical limit of the phenomenon we first regularise the discontinuity in the potential by taking a continuous linearly rising function over a distance a. We show that the correct classical limit, which is when no reflection occurs, is obtained for large values of a, not for vanishing a. The latter actually corresponds to a purely quantum case, albeit in a limiting case of a discontinuously abrupt change in the potential at a single point in space.
In this work we propose an alternative procedure to calculate the classical limit ( ® 0) of the quantum mechanical equations of motion. We compare our method with the Ehrenfest theorem, the WKB approximation and the Bohm quantum potential Q ® 0. We also translate an Einstein's paper about the relation between quantum and classical theories.
The classical limit problem of quantum mechanics is a particular example of a broader issue concerning the relationship between two physical theories, one empirically valid in a subdomain of the other. It is a basic requirement that a theory superseding the previous one must be compatible with the former at a suitable limit. As a prelude to discussing the classical limit of quantum mechanics, we recall some relevant aspects of this general issue.
The subject of this paper are quantum-informational bases and frontiers of psychosomatic integrative medicine, oriented to the healing of person as a whole and not diseases as symptoms of disorders of this wholeness, implying their macroscopic quantum origin. In the focus of the related quantum-holistic methods are acupuncture system and consciousness - which as macroscopic quantum biosystems have (fundamental!) quantum-informational structure of Hopfield-like holographic neural network, within the Feynman propagator version of quantum mechanics. In this context, all acupuncture-based and consciousness-based approaches and techniques can be essentially considered as quantum-informational therapies, via bioresonant imposing of new boundary conditions in the energy-state space of the acupuncture system/ consciousness, with downward quantum-holographic biofeedback influence on the cells conformational protein changes and genes expression. This might provide fundamental quantum-informational framework for better understanding of the nature of psychosomatic diseases as well as limitations of the healing methods, which might help in developing strategies for integrative medicine.
In this work we propose an alternative procedure to calculate the classical limit (® 0) of the quantum mechanical equations of motion. We compare our method with the Ehrenfest theorem, the WKB approximation and the Bohm quantum potential Q ® 0. We also translate an Einstein's paper about the relation between quantum and classical theories.
Expressing the Wigner distribution function in Dira c notation reveals its resemblance to a classical trajectory in phase space.
Although most people associate the correspondence principle with quantum mechanics, it has been used more generally for many years in the development of theories in essentially all branches of physics. Based upon simple assumptions which would seem to be necessary for any exact science, a more precise statement of the generalized correspondence principle is provided, with due regard to the accuracy of theories. Term correspondence is presented as a corollary to the principle. Evidently, the principle has as strong a basis as other scientific theories. Its value in theory development as well as in furthering the understanding of the history of science is discussed.
Definitions of the momentum density, energy density, and densities of some other physical quantities are added to the standard Klein--Gordon theory. The densities introduced allow calculation of momentum and energy and their uncertainties of a localised Klein--Gordon field. It is shown that at certain conditions the momentum, energy, and potential energy of a localised Klein--Gordon field satisfy the relativistic Hamilton--Jacobi equation.
Becausethe information contained in a wave function is not enough to describe an individual particle. the suggestion to define
the state of a particle by two-wave functions is used. In the double-wave description. a general formula telling the traversal
time of a particle in tunneling is obtained. When the double-wave description over an initial parameter is averaged. it redurn
to the usual description. Two new relations Letween the tmnsmission and the reflected cwfficients are fwnded in the new cescription.
A complete system of states, closely related to the coherent states, and which incorporates the necessary algebra of fermion states, is used to derive an expression for the corresponding propagator via functional integrals. The calculation, based on a product expansion of the evolution operator, leads to an exact Feynman propagator from which the expectation value of the density matrix is obtained.
Instead of regarding the classical limit as the limit ℏ-->0, an alternative view based on the physical interpretation of the elements of the density matrix is proposed. The derivation of the classical limit of quantum mechanics is carried out in two stages. First, the statistical classical limit is derived, and then, with an appropriate initial condition, the deterministic classical limit is obtained. The derivation hinges on the use of the Feynman path-integral formulation of quantum mechanics in terms of the density matrix. It is shown that deterministic (Hamilton's or Newton's) classical mehanics is not the classical limit of Schrödinger's wave mechanics. The classical limit of Schrödinger's wave mechanics is only statistical and corresponds to the classical Liouville equation. In order to obtain the deterministic classical limit, it is necessary to start out initially with a quantum mechanical mixture.
Using a procedure based on the limit , we show that the classical limiting method based on the Bohm quantum potential (Q 0) is not necessary to characterize the classical limit of quantum mechanics. PACS No.: 03.65.Ta
Using a symplectic notation which allows the equations to apply equally to a wavepacket representing a spin-less quantum particle or to a cluster of identical classical particles without mutual interactions, the study of the time-evolution of moments is extended to arbitrary orders for arbitrary Hamiltonians. Appropriately symmetrised moments for the quantum case are seen to play a special role and correspond very closely to the classical moments.
In the present paper we propose a method of quantization of mechanical systems that is applicable, inter alia, to systems acted on by nonconservative forces, such as dissipative systems. First, the correspondence between classical and quantum mechanics is considered and we make some important observations that can be used in defining the rules of quantization. Next, the quantization rules are presented and we give a few examples of the application of these rules to mechanical systems that are difficult to quantize within any other systemetic approach. Finally, it is shown that for any quantum-mechanical system there are stationary solutions of the evolution equations, and that the existence of these solutions is directly related to the proposed quantization rules.
The Web and its tools of orientation: how better to understand the information available on the Internet by the analysis of references? In the incessant and chaotic frenzy of the Internet, suitable tools and techniques are urgently required for use by information professionals. The Internet has become a leading venue for communication among researchers, both scientists and other scholars. The role of mediation for these professionals has never been so important, owing to the actual deficiencies of the search systems on the Internet. This article expounds the tools that are now available for the identification and evaluation of Internet sources. After a discussion of the techniques of discrimination and cartography, a method of constructing a navigation map of the Web by the analysis of citations is explained.
This brief comprehensive review presents the recent development in basic and applied science and engineering of finely dispersed
particles and related systems in general, but more profound and in-depth treatise are related to the liquid-liquid finely
dispersed systems i.e. emulsions and double emulsions. The electro-viscoelastic behavior of e.g., liquid/liquid interfaces
(emulsions and double emulsions) is based on three forms of “instabilities”; these are rigid, elastic, and plastic. The events
are understood as interactions between the internal (immanent) and external (incident) periodical physical fields. Since the
events at the interfaces of finely dispersed systems have to be considered at the molecular, atomic, and/or entities level
it is inevitable to introduce the electron transfer phenomenon beside the classical heat, mass, and momentum transfer phenomena
commonly used in chemical engineering. Three possible mathematical formalisms have been derived and discussed related to this
physical formalism, i.e. to the developed theory of electroviscoelasticity. The first is stretching tensor model, where the
normal and tangential forces are considered, only in mathematical formalism, regardless to their origin (mechanical and/or
electrical). The second is classical integer order van der Pol derivative model. Finally, the third model comprise an effort
to generalize the previous van der Pol differential equations, both, linear and nonlinear; where the ordinary time derivatives
and integrals are replaced by corresponding fractional-order time derivatives and integrals of order p < 2 (p = n − δ, n = 1, 2, δ ≪ 1). In order to justify and corroborate more general approach the obtained calculated results were compared to
those experimentally measured using the representative liquid-liquid system. Also, a new idea related to the probable discussion
and/or elucidation of the problems in the theoretical and experimental status of decoherence is mentioned.
In the context of the general problem of equivalence between classical mechanics and quantum mechanics in the macroscopic limit, we point out that, for the particular case of the one-dimensional Coulomb potential, the quantum-mechanical result in the classical limit, corresponding to a certain superposition of odd- and even-parity energy eigenfunctions, leads to inconsistency with classical mechanics. It is shown that the contradiction persists even if the singularity of the Coulomb potential is treated as the limiting case of a modified Coulomb potential in which the singularity has been smoothed out. The possible implication of this paradoxical finding is briefly discussed.
This is a short review of Bohmian Mechanics with special emphasis on the role of probability within this deterministic quantum
theory without observers. I discuss the equations of motion and the statistical mechanics of this theory.
The recent development in basic and applied science and engineering of finely dispersed systems is presented in general, but more attention has been paid to the liquid-liquid finely dispersed systems or to the particular emulsions and double emulsions. The selected systems for theoretical and experimental research were emulsions and double emulsions that appeared in the pilot plant for extraction of uranium from wet phosphoric acid. The objective of this research was to try to provide a new or different approach to elaborate the complex phenomena that occur at developed liquid-liquid interfaces. New concepts were introduced, the first is a concept of an entity, and the corresponding classification of finely dispersed systems and the second concept consider the introduction of an almost forgotten basic electrodynamics element memristor, and the corresponding memristive systems. Based on these concepts a theory of electroviscoelasticity was proposed and experimentally corroborated using the selected representative liquid-liquid system. Also, it is shown that the droplet, and/or droplet-film structure, that is, selected emulsion and/or double emulsion may be considered as the particular example of memristive systems.
Expressing the Wigner distribution function in Dirac notation reveals its resemblance to a classical trajectory in phase space. Comment: 3 pages, 1 figure
We consider the arrival time distribution defined through the quantum probability current for a Gaussian wave packet representing free particles in quantum mechanics in order to explore the issue of the classical limit of arrival time. We formulate the classical analogue of the arrival time distribution for an ensemble of free particles represented by a phase space distribution function evolving under the classical Liouville's equation. The classical probability current so constructed matches with the quantum probability current in the limit of minimum uncertainty. Further, it is possible to show in general that smooth transitions from the quantum mechanical probability current and the mean arrival time to their respective classical values are obtained in the limit of large mass of the particles. Comment: Latex, 10 pages, 3 eps figures; analysis carried out with a more general wave function; comments and references added
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 m 0 = ħc/γ 1 2 ≈ 10 −5 g . This nonlinear equation may help one to understand the process of measurement of a quantum-mechanical system by means of a classical measuring instrument.
The Statistical Interpretation of quantum theory is formulated for the purpose of providing a sound interpretation using a minimum of assumptions. Several arguments are advanced in favor of considering the quantum state description to apply only to an ensemble of similarily prepared systems, rather than supposing, as is often done, that it exhaustively represents an individual physical system. Most of the problems associated with the quantum theory of measurement are artifacts of the attempt to maintain the latter interpretation. The introduction of hidden variables to determine the outcome of individual events is fully compatible with the statistical predictions of quantum theory. However, a theorem due to Bell seems to require that any such hidden-variable theory which reproduces all of quantum mechanics exactly (i.e., not merely in some limiting case) must possess a rather pathological character with respect to correlated, but spacially separated, systems.
We prove that a wave-packet solution in the WKB approximation to the Schrödinger equation follows the classical trajectory, and we discuss the nature of this solution.
Es wird die Bewegung eines Teilchens auf einer geradlinigen Strecke betrachtet, an deren Enden es ohne Energieverlust reflektiert wird; die Anfangslage und Anfangsgeschwindigkeit sind so scharf gegeben, wie es die Unschrferelation erlaubt. Einfache, schnell konvergente Reihen sind fr zwei Grenzflle bekannt; 1. das Teilchen ist noch angenhert lokalisierbar (t klein); 2. das Teilchen ist nicht mehr lokalisierbar, die Bewegung wird durch Superposition von Eigenschwingungen beschrieben (t gro). Im folgenden wird gezeigt, da mit Hilfe einer-Transformation eine exakte Formel aufgestellt werden kann, welche fr alle Zeitent gilt und sich in den Grenzfllen (t klein,t gro) auf die bekannten Ausdrcke reduziert.