Particles classical view

If the terms, particles or waves, imply classical physics systems, as they do when invoking the principle of complementarity, then the results analyzed here imply the particle-wave duality belonging to a classical view with no place to claim in QM as presented here. 

The interpretation of QM formalism in terms of the complementarity principle leads to puzzling situations the particle-wave view seems to be fundamentally flawed to the extent classical concepts do not belong to an interpretive framework to quantum mechanics. 

Does the particle nature of light cause its wave aspects Or vice versa All these questions may only be asked from the point of view of classical physics, they only have meaning from the classical view. Once quantum mechanical physics enters the scene, no one even attempts to answer the questions on the classical level, if my guess that brain and mind are parallel aspects of a more fundamental reality is nebulous, perhaps it will take on some relevance when a "quantum mechanics of philosophy" will be available, whether a process of mind studying mind will accomplish such a feat is still an open question. 

For an asymmetric reaction, the reaction coordinate at the TS in the traditional view includes both proton and heavy particle classical motions consistent with the Hammond postulate [15], the TS becomes more geometrically similar to the product as the reaction becomes more endothermic, and more similar to the reactant as it becomes more exothermic. Thus the transverse vibration at the TS -whose ZPE is relevant for the rate - more and more involves the proton motion, and in either limit approaches the bound proton stretch vibration of the product BH or the reactant AH. This effect decreases the KIE, resulting in a kH/ o vs. AGi xn trend which is maximal at AGj xn O drops off as the reaction becomes... 

For a brief introduction, we refer again to Fig. 25.5(a), which immediately reveals that it requires an activation energy of for an atom (or a molecule) to be transferred from a site A to another, geometrically identical site B on the same periodic surface. In the classical view, this two-dimensional diffusion process can be thought of as a sequence of individual and statistical hopping events of frequency v, each activated with an energy as pointed out, for example, by Roberts and McKee [35]. The inverse of this frequency then yields the residence time, t, of the particle in the respective site. For thermally equilibrated particles, the temperature dependence of the classical surface diffusion is described by the well-known Arrhenius relation... 

In the classical view of the world, a moving particle has a definite location at any instant, whereas a wave is spread out in space. If an electron has the properties of both a particle and a wave, what can we determine about its position in the atom In 1927, the German physicist Werner Heisenberg postulated the uncertainty principle, which states that it is impossible to know the exact position and momentum (mass times speed) of a particle simultaneously. For a particle with constant mass m, the principle is expressed mathematically as... 

During the 19th century, a number of experimental observations were made which were not consistent with the classical view that matter could interact with energy in a continuous form. Work by Einstein, Planck and Bohr indicated that in many ways electromagnetic radiation could be regarded as a stream of particles (or quanta), for which the energy, E, is given by the Bohr equation, as follows ... 

If two systems are in equilibrium, thus in thermal equilibrium, they exhibit equal temperatures. On the other hand, a collection of particles are in equilibrium, if there is a certain velocity distribution. The velocity can serve as a measure of the temperature of a particle. If we break down the collection of particles to thermodynamic systems, the concept of thermal equilibrium is violated. According to the classical view of thermodynamics, we could increase the entropy if we pick out two systems of different temperatures and allow equilibrating. Obviously, the application of thermodynamics is restricted to entities that consist of more than one particle. Actually, the variables in classical thermodynamics rely on averages of large ensembles. 

Classical view Identical distinguishable Indistinguishable particles Indistinguishable particles... 

Let s get back to the photon. In a beam of red light with frequency V = 5.0 10 s for example, an atom can absorb only hv = 3.3 10 J of energy at a time—not 1.1 10 J or 3.3 10 J or any other amount. This violates the classical view of radiation, in which the energy of a beam of light is a continuous variable. It suggests instead that radiation has some characteristics traditionally associated with particles—that it behaves like separate chunks of energy rather than a single, continuous wave. 

We base our quantum version of radiation on one relationship, Planck s law (Eq. 1.2) Ephoton The quantum mechanics of matter also grow out of a single equation, one that ascribes a property to matter completely alien to the classical view a wavelength. At the tiny distances over which atoms and subatomic particles interact, matter exhibits the characteristics of waves, consistent with a de Broglie wavelength. 

In its basic expression, the Drude model does not predict that the absorption bandwidth is affected by particle size. Experimentally, colloidal systems having a weak cluster-matrix interaction show a well-established inverse correlation with respect to the plasmon bandwidth with particle size. In order to describe the bandwidth dependency on particle size. Hovel et al. [47] proposed a classical view of free-electron metals here, the scattering of electrons with other electrons, phonons, lattice defects and impurities leads to a damping of the Mie resonance. Briefly, in realistic metals, the dielectric function is composed of contributions from both interband transitions and the free-electron portion [48]. The free-electron dielectric function can be modified by the Dmde model to account for this dependency, giving [47-50]... 

It is convenient to analyse tliese rate equations from a dynamical systems point of view similar to tliat used in classical mechanics where one follows tire trajectories of particles in phase space. For tire chemical rate law (C3.6.2) tire phase space , conventionally denoted by F, is -dimensional and tire chemical concentrations, CpC2,- are taken as ortliogonal coordinates of F, ratlier tlian tire particle positions and velocities used as tire coordinates in mechanics. In analogy to classical mechanical systems, as tire concentrations evolve in time tliey will trace out a trajectory in F. Since tire velocity functions in tire system of ODEs (C3.6.2) do not depend explicitly on time, a given initial condition in F will always produce tire same trajectory. The vector R of velocity functions in (C3.6.2) defines a phase-space (or trajectory) flow and in it is often convenient to tliink of tliese ODEs as describing tire motion of a fluid in F with velocity field/ (c p). 

Quantization of the Electromagnetic Field.—Instead of proceeding as in the previous discussion of spin 0 and spin particles, we shall here adopt essentially the opposite point of view. Namely, instead of formulating the quantum theory of a system of many photons in terms of operators and showing the equivalence of this formalism to the imposition of quantum rules on classical electrodynamics, we shall take as our point of departure certain commutation rules which we assume the field operators to satisfy. We shall then show that a... 

Another example of slight conceptual inaccuracy is given by the Wigner function(12) and Feynman path integral(13). Both are useful ways to look at the wave function. However, because of the prominence of classical particles in these concepts, they suggest the view that QM is a variant of statistical mechanics and that it is a theory built on top of NM. This is unfortunate, since one wants to convey the notion that NM can be recovered as an integral part of QM pertaining to for macroscopic systems. 

This quantity is of great importance, since it actually contains all information about electron correlation, as we will see presently. Like the density, the pair density is also a non-negative quantity. It is symmetric in the coordinates and normalized to the total number of non-distinct pairs, i. e., N(N-l).8 Obviously, if electrons were identical, classical particles that do not interact at all, such as for example billiard balls of one color, the probability of finding one electron at a particular point of coordinate-spin space would be completely independent of the position and spin of the second electron. Since in our model we view electrons as idealized mass points with no volume, this would even include the possibility that both electrons are simultaneously found in the same volume element. In this case the pair density would reduce to a simple product of the individual probabilities, i.e.,... 

In physical chemistry the most important application of the probability arguments developed above is in the area of statistical mechanics, and in particular, in statistical thermodynamics. This subject supplies the basic connection between a microscopic model of a system and its macroscopic description. The latter point of view is of course based on the results of experimental measurements (necessarily carried out in each experiment on a very large number of particle ) which provide the basis of classical thermodynamics. With the aid of a simple example, an effort now be made to establish a connection between the microscopic and macroscopic points of view. 

Werner Heisenberg (1901-1976 Nobel Prize for physics 1932) developed quantum mechanics, which allowed an accurate description of the atom. Together with his teacher and friend Niels Bohr, he elaborated the consequences in the "Copenhagen Interpretation" — a new world view. He found that the classical laws of physics are not valid at the atomic level. Coincidence and probability replaced cause and effect. According to the Heisenberg Uncertainty Principle, the location and momentum of atomic particles cannot be determined simultaneously. If the value of one is measured, the other is necessarily changed. 

This is a recurrent theme of quantum theory. Many quantum systems can be formulated exactly in terms of a wave equation and the behaviour of the system will be described exactly by the wavefunction, the solution to the wave equation. What is not always appreciated is that this is a mathematical description only, which does not ensure understanding of the event in terms of a comprehensible physical model. The problem lies therein that the description is only possible in terms of a wave formalism. Understanding of the physical behaviour however, requires reduction to a particle model. The wave description is no more than a statistically averaged picture of the behaviour of many particles, none of which follows the actual statistically predicted course. The wave description is non-classical, and the particle model is classical. Mechanistic understanding is possible only in terms of the classical approach, and a mathematically precise description only in terms of the wave formalism. The challenge of quantum theory is to reconcile the two points of view. 

In the early development of the atomic model scientists initially thought that, they could define the sub-atomic particles by the laws of classical physics—that is, they were tiny bits of matter. However, they later discovered that this particle view of the atom could not explain many of the observations that scientists were making. About this time, a model (the quantum mechanical model) that attributed the properties of both matter and waves to particles began to gain favor. This model described the behavior of electrons in terms of waves (electromagnetic radiation). 

Niels Bohr, a pioneer in both disciplines, emphasized the significance of classical vj. quantal arguments in particle penetration. Not the least in view of the complexity of ab initio computations in this area, such considerations keep being relevant. This note adds new points to an old discussion based on recent developments. 

Parallel with the phenomenological development, an alternative point of view has developed toward thermodynamics, a statistical-mechanical approach. Its philosophy is more axiomatic and deductive than phenomenological. The kinetic theory of gases naturally led to attempts to derive equations describing the behavior of matter in bulk from the laws of mechanics (first classic, then quanmm) applied to molecular particles. As the number of molecules is so great, a detailed treatment of the mechanical problem presents insurmountable mathematical difficulties, and statistical methods are used to derive average properties of the assembly of molecules and of the system as a whole. 

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