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Significance of Ehrenfesttheorem in Quantum-Classical relationship
Abstract
The significance of Ehrenfest Theorem in quantum-classical relationship
is discussed in terms of the general formulation of the theorem. With the
ensemble interpretation of the quantum mechanical -function the
Generalised Ehrenfest Theorem reveals some interesting exact
relationships between quantum- and classical expectation values. These general
results seem to imply a deep rooted unity (inspite of apparent radical
differences in conceptual structures) between classical and quantum mechanics.
Some significant consequences and important physical insights which follow
from the general formulation are discussed with examples. Most important
is that it offers, under reasonable approximations, a pure quantum mechanical
description of the Stern-Gerlach experiment with realistic inhomogeneous
magnetic field .
... The conditions are complete states given by open subsets of the system's smooth state space E S (A), a proper subset of the standard state space, see §4.2. The qr-numbers are sections of the sheaf of Dedekind reals R D (E S (A)) in Shv(E S (A)), the topos of sheaves on E S (A), see §4. 3. ...
... The expectation values of standard quantum mechanics are shown to be interpretable as infinitesimal qr-numbers in §3.3. The relationship between the equations of motion for infinitesimal qr-numbers and the classical equations of motion is different from that given in Ehrenfest's theorem [3] which interprets the expectation values of the particle's position and momentum as standard real numbers. The proof that the equation of motion for the infinitesimal qr-number values of the particle's position and momentum are equivalent to the standard quantum mechanical equations of motion depends upon the position operators, {Q j } 3 j=1 , having only continuous spectrum σ c (Q j = R. ...
The Heisenberg equations of motion for a quantum particle of mass m are deduced from the infinitesimal qr-number equations of motion for the particle. The infinitesimal qr-number equations, and hence the standard quantum mechanical equations, are related to the qr-number equations in much the same way as the equations of geometric optics are related to those of wave optics. The qr-number equations of motion for a quantum particle of mass m describe the motion of a lump, given by on open set in the qr-number space of the particle, while the infinitesimal qr-number equations describe the motion of a point-like particle. The qr-number equations of motion are the Hamiltonian equations of motion for a classical particle of mass m expressed in qr-numbers. The proof requires that the particle's position operators have only continuous spectrum and the force functions are smooth.
... Using Parseval's formula and the fact that both ρ(x, t) and w(p, t) are normalized to unity, we find that the integral of h(p, t) has to vanish too. Using the continuity equation (7) and the first statistical condition (2), we found two results (namely (18) and (21)), which reduce for h(p, t) = 0 to characteristic relations of the quantum mechanical formalism. However, the function h(p, t), as well as the probability density w(p, t) we are finally interested in, is still unknown, because the validity of the deterministic relation (8) is not guaranteed in the present general formalism allowing for type 3 theories. ...
... Thus, quantum mechanics belongs to the class of theories defined by the above conditions. We see that the statistical conditions (2), (3) comprise both quantum mechanical and classical statistical theories; these relations express a "deep-rooted unity" [18] of the classical and quantum mechanical domain of physics. We found an infinite number of statistical theories which are all compatible with our basic conditions and are all on equal footing so far. ...
It is shown that Schrödinger's equation may be derived from three postulates. The first is a kind
of statistical metamorphosis of classical mechanics, a set of two relations which are obtained from
the canonical equations of particle mechanics by replacing all observables by statistical averages.
The second is a local conservation law of probability with a probability current which takes the
form of a gradient. The third is a principle of maximal disorder as realized by the requirement of
minimal Fisher information. The rule for calculating expectation values is obtained from a fourth
postulate, the requirement of energy conservation in the mean. The fact that all these basic relations
of quantum theory may be derived from premises which are statistical in character is interpreted as
a strong argument in favor of the statistical interpretation of quantum mechanics. The structures of
quantum theory and classical statistical theories are compared, and some fundamental differences are
identified.
... 6 The Ehrenfest theorem links quantum mechanics to classical principles, asserting that the predicted values in quantum mechanics align with classical principles. 7 The microscopic fluid's resistance to flow is attributed to intermolecular potential among particles, which can be explained using quantum mechanics to construct a viscosity relation. 8 Numerous researchers have developed models for precise viscosity calculations and predictions in engineering, leaving outstanding theoretical and molecular perspectives in the literature. ...
This study extends the recently introduced theory for predicting fluid viscosity by focusing on the thermodynamic dimension (DT) and its relationship to the effective intermolecular potential, Ueff (r, T). The DT describes the degree of freedom in fluid–particle interactions, ranging from three in solids to zero in perfect gases (0 ≤ DT ≤ 3). The DT of fluid varies between these limits depending on temperature and pressure. Unlike traditional models, this method views a fluid as a mix of free particles and temporary clusters (t-clusters), with viscosity resulting from gaseous and solid-like interactions. Viscosity is the sum of dilute gas viscosity and viscosity caused by intermolecular interactions. The theory uses explicit thermodynamic relations to link the Ueff (r, T) to the fluid's equation of state (EoS), giving a unified approach that connects the viscosity equation to the EoS. Argon fluid serves as a case study to demonstrate the model's correctness. The model accurately predicts viscosity throughout a temperature range of 100–1000 K and pressures up to 1000 MPa, with average absolute relative deviations less than 2%. The model's accuracy and computational efficiency enable it to be applied to other simple fluids while upholding the corresponding states principle. These results demonstrate how this theory may be used to unify quantum and classical thermodynamics viewpoints, providing a new framework for understanding and predicting fluid behavior under a variety of circumstances. Because of its accuracy and computational simplicity, the method shows promise for use in fluid mechanics and thermophysical modeling.
... Our theoretical tool is the set of coupled Ehrenfest equations for the translational and spin degrees of freedom [8,9]. They are derived from the Schrödinger-Pauli equation for a neutral particle endowed with the magnetic moment µ = gs, where g is the gyromagnetic ratio and the spin vector s is built from the appropriate spin matrices. ...
The motion of a neutral atom endowed with a magnetic moment interacting with the magnetic field is determined from the Ehrenfest-like equations of motion. These equations for the average values of the translational and spin degrees of freedom are derived from the Schr\"odinger-Pauli wave equation and they form a set of nine coupled nonlinear evolution equations. The numerical and analytic solutions of these equations are obtained for the combination of the rotating magnetic field of a wave carrying orbital angular momentum and a static magnetic field. The running wave traps the atom only in the transverse direction while the standing wave traps the atom also in the direction of the beam.
... The Drude-Lorentz model's predictions are completely supported by the classical oscillator model as well as quantum mechanical features like electronic dipole moments of materials. To justify the microscopic qualities exhibited by classical and quantum techniques using this model, it is necessary to understand the Ehrenfest theorem which shows that quantum mechanical predicted values fundamentally follow classical mechanical conditions [31]. The Kohn-Sham approach is praised for its ease of use in relating a many-body system to a non-interacting system and thereby solving it using Kohn-Sham density functional theory [32]. ...
Plasmonics is a technologically advanced term in condensed matter physics that describes surface plasmon resonance where surface plasmons are collective electron oscillations confined at the dielectric-metal interface and these collective excitations exhibit profound plasmonic properties in conjunction with light interaction. Surface plasmons are based on nanomaterials and their structures; therefore, semiconductors, metals, and two-dimensional (2D) nanomaterials exhibit distinct plasmonic effects due to unique confinements. Recent technical breakthroughs in characterization and material manufacturing of two-dimensional ultra-thin materials have piqued the interest of the materials industry because of their extraordinary plasmonic enhanced characteristics. The 2D plasmonic materials have great potential for photonic and optoelectronic device applications owing to their ultra-thin and strong light-emission characteristics, such as; photovoltaics, transparent electrodes, and photodetectors. Also, the light-driven reactions of 2D plasmonic materials are environmentally benign and climate-friendly for future energy generations which makes them extremely appealing for energy applications. This chapter is aimed to cover recent advances in plasmonic 2D materials (graphene, graphene oxides, hexagonal boron nitride, pnictogens, MXenes, metal oxides, and non-metals) as well as their potential for applied applications, and is divided into several sections to elaborate recent theoretical and experimental developments along with potential in photonics and energy storage industries.
... On the semiclassical level, i.e. when we deal with a modified dynamics due to a redefined conjugate momentum, we provided a detailed study of the resulting dynamics in all the three sets of variables. This study is justified by the so-called Ehrenfest theorem [25], since the modified classical solutions well-approximate the behavior of the quantum mean values when the full quantum polymer approach is implemented. ...
We analyze the Bianchi I cosmology in presence of a massless scalar field and describe its dynamics via a semiclassical and quantum polymer approach. We investigate the morphology of the emerging Big Bounce by adopting three different sets of configurational variables: the Ashtekar connections, the Universe volume plus two anisotropy coordinates and a set of anisotropic volume-like coordinates. In the semiclassical analysis we demonstrate that the Big Bounce has a universal nature in both the two sets of volume-like variables, i.e. its critical energy density has a maximum value fixed by fundamental constants only. On the contrary, when adopting the Ashtekar connections (the privileged variables as dictated by Loop Quantum Gravity) the value of the critical density depends on the Cauchy problem for the dynamics and specifically on the conjugate momentum to the scalar field, which is a constant of motion in the present analysis. Also, a cosmological constant is included in the Ashtekar connections' formulation and some interesting results are mentioned making a comparison between the synchronous dynamics and that one when the scalar field is taken as a relational time. From a pure quantum point of view, we investigate the Bianchi I dynamics only in terms of the Ashtekar connections and we demonstrate that the Wheeler-DeWitt equation is not an appropriate quantum polymer procedure, since its associated probability density is not positive for a wave packet, even after the frequency separation procedure is performed. Thereby, we apply the Arnowitt-Deser-Misner (ADM) reduction of the variational principle and then we quantize the system. We study the resulting Schr\"{o}dinger dynamics, stressing that the wave packet peak behavior over time singles out common features with the semiclassical trajectory, confirming the non-universal character of the emerging Big Bounce also on a quantum level.
... Our theoretical tool is the set of coupled Ehrenfest equations for the translational and spin degrees of freedom [8,9]. They are derived from the Schrödinger-Pauli equation for a neutral particle endowed with the magnetic moment µ = gs, where g is the gyromagnetic ratio and the spin vector s is built from the appropriate spin matrices. ...
The motion of a neutral atom endowed with a magnetic moment interacting with the magnetic field is determined from the Ehrenfest-like equations of motion. These equations for the average values of the translational and spin degrees of freedom are derived from the Schr\"odinger-Pauli wave equation and they form a set of nine coupled nonlinear evolution equations. The numerical and analytic solutions of these equations are obtained for the combination of the rotating magnetic field of a wave carrying orbital angular momentum and a static magnetic field. The running wave traps the atom only in the transverse direction while the standing wave traps the atom also in the direction of the beam.
This work presents a different perspective on viscosity and a new interpretation of it by treating the fluid as a fractal lattice incorporating temporary molecular clusters (t-clusters) and using the thermodynamic dimension (DT) idea. The DT connects fluid viscosity to its EoS by computing the effective intermolecular potential, U(r, T), which may be found via thermodynamic relations from the fluid equation of state (EoS). Finally, a general viscosity equation is developed utilizing standard and well-established statistical thermodynamic relations. The approach is applied to nitrogen fluid as a case study because of the fluid's extensiveness data in the literature. Less than 2% is the absolute average deviation percent (AAD%) for viscosity prediction in the range of 65 K to 1000 K for pressures less than 2000 MPa. It is simple to code the viscosity equation that is obtained.
The pseudo-PT symmetric Dirac equation is proposed and analyzed by using a non-unitary Foldy-Wouthuysen transformations. A new spin operator PT symmetric expectation value (called the mean spin operator) for an electron interacting with a time-dependent electromagnetic field is obtained. We show that spin magnetization - which is the quantity usually measured experimentally - is not described by the standard spin operator but by this new mean spin operator to properly describe magnetization dynamics in ferromagnetic materials and the corresponding equation of motion is compatible with the phenomenological model of the Landau-Lifshitz-Gilbert equation (LLG).
The classical behaviour of a quantum ensemble is expected to be achievable only by a limiting procedure. In this communication we discuss two remarkable examples in which position and momentum probability density functions using an arbitrary Gaussian ensemble (described by a non-coherent Gaussian wave packet) develop identically both in classical and quantum mechanics. The first case is propagation in free space and the second in a harmonic oscillator potential. In both cases quantum potential and quantum force (defined in Bohm's theory) are non-zero. This may be stated to be the outcome of the present investigation.
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.
This elementary review paper presents the compelling evidence which supports the notion that quantum mechanics is much too simple a theory to adequately describe a complex world. Rigorous arguments based on algorithmic complexity theory are used to show that both the quantum Arnol'd cat and a broad category of finite, bounded, undriven, quantum systems do not obey the correspondence principle, implying that quantum mechanics is also not complete. An experiment, well within current laboratory capability, is proposed which can expose the inability of quantum mechanics to adequately describe macroscopic chaos. In its final section, this paper describes a theoretical framework that provides a proper setting for interpreting these surprising results.
A motion of a Gaussian wave packet of a single electron in a harmonic potential is obtained by solving the time dependent Schrodinger equation analytically. The analytic formula of the phase of the wave function which depends on both time and space is also presented explicitly. The probability density of the wave packet moves changing its width and central position periodically. The period of the change of the width is half of that of the central position. The interpretation of these periodic changes in terms of the Green's function which can be derived by the path integration formulation is also discussed.
The equations of motion for phase-space moments and correlations are derived systematically for quantum and classical dynamics, and are solved numerically for chaotic and regular motions of the Hénon-Heiles model. For very narrow probability distributions, Ehrenfest’s theorem implies that the centroid of the quantum state will approximately follow a classical trajectory. But the error in Ehrenfest’s theorem does not scale with ħ, and is found to be governed essentially by classical quantities. The difference between the centroids of the quantum and classical probability distributions, and the difference between the variances of those distributions, scale as ħ2, and so are the true measures of quantum effects. For chaotic motions, these differences between quantum and classical motions grow exponentially, with a larger exponent than does the variance of the distributions. For regular motions, the variance of the distributions grows as t2, whereas the differences between the quantum and classical motions grow as t3.
The classical limit of quantum mechanics is usually discussed in terms of Ehrenfest’s theorem, which states that, for a sufficiently narrow wave packet, the mean position in the quantum state will follow a classical trajectory. We show, however, that that criterion is neither necessary nor sufficient to identify the classical regime. Generally speaking, the classical limit of a quantum state is not a single classical orbit, but an ensemble of orbits. The failure of the mean position in the quantum state to follow a classical orbit often merely reflects the fact that the centroid of a classical ensemble need not follow a classical orbit. A quantum state may behave essentially classically, even when Ehrenfest’s theorem does not apply, if it yields agreement with the results calculated from the Liouville equation for a classical ensemble. We illustrate this fact with examples that include both regular and chaotic classical motions.
Theory reduction is analyzed and examples are presented from various branches of physics. The procedure takes different forms in different theories. Examples from various theories are arranged in increasing order of difficulty. Special emphasis is placed on the quantum to classical reduction. It is argued that there is good and interesting physics in theory reduction and that it deserves more attention than it has been receiving in the past.
Aus der Schrdingerschen Gleichung lt sich durch eine kurze elementare Rechnung ohne Veruachlssigung die Beziehung
m\fracd2 dt2 \smallint \smallint \smallint dt.YY* .x = \smallint \smallint \smallint dt.YY* ( - \frac¶V¶x )m\frac{{d^2 }}{{dt^2 }}\smallint \smallint \smallint d\tau .\Psi \Psi * .x = \smallint \smallint \smallint d\tau .\Psi \Psi * \left( { - \frac{{\partial V}}{{\partial x}}} \right)
ableiten, die fr ein kleines und klein bleibendes Wellenpaket (m von der Ordnung 1 g) besagt, da die Beschleunigung seiner Lagekoordinaten im Sinne der Newtonschen Bewegungsgleichungen zur rtlichen Kraft -V/x pat.