All results of classical mechanics of one particle in three dimensions presented so far do directly transform to the situation of N particles. For a system of N [Pg.21]

The classic mechanism of particles being lifted up through the bed in the bubble wake and in the spout behind a bubble (see Chapter 1, p 18) still operates when the bed is composed of two distinct layers jetsam at the bottom and flotsam above. However, Rowe, Nienow and co-workers also showed that bubbles are responsible for segregation [Pg.69]

Classical Mechanics of One-Particle Angular Momentum. Consider a moving particle of mass m. We set up a Cartesian coordinate system that is fixed in space. Let r be the vector from the origin to the instantaneous position of the particle. We have [Pg.102]

A simple realization of the harmonic oscillator in classical mechanics is a particle which is acted upon by a restoring force proportional to its displacement from its equihbrium position. Considering motion in one dimension, this means [Pg.201]

All of these mechanisms which affect crosslink density were confirmed by experimental studies. The classic case of a reactive particle filler is silica filled polysiloxane (Figure 6.25). Silica particles have numerous OH groups which react with the crosslinking component of polysiloxane. Modification of silica by silanes reduces reinforcement. [Pg.338]

In classical mechanics the kinetic energy of a particle of the form [Pg.92]

In classical mechanics, the kinetic energy of a particle is defined asT = mi. Use results from Section 1.4 to show that, for a particle moving vertically in the earth s gravitational field (with g assumed constant), T + V = mvQ + mgx, so 7 + V is constant. [Pg.19]

At last, we can resolve the paradox between de Broglie waves and classical orbits, which started our discussion of indeterminacy. The indeterminacy principle places a fundamental limit on the precision with which the position and momentum of a particle can be known simultaneously. It has profound significance for how we think about the motion of particles. According to classical physics, the position and momentum are fully known simultaneously indeed, we must know both to describe the classical trajectory of a particle. The indeterminacy principle forces us to abandon the classical concepts of trajectory and orbit. The most detailed information we can possibly know is the statistical spread in position and momentum allowed by the indeterminacy principle. In quantum mechanics, we think not about particle trajectories, but rather about the probability distribution for finding the particle at a specific location. [Pg.140]

We start by considering the relativistic classical mechanics of a particle in free space. As we have already seen in chapter 2, the momentum P of the particle is given by [Pg.74]

Lagrangian mechanics is a way of writing the classical mechanics of Newton in a way that has the same form in any coordinate system. It is convenient for problems in which Cartesian coordinates cannot conveniently be used. We specify the positions of the particles in a system by the coordinates , 3,..., qn, where n is the number of [Pg.1270]

The L term contains all the dependence on the angular motion of the electron. In classical mechanics, we express the kinetic energy of a particle as wjv /2, and we can break the contributions to the velocity up into a radial speed and an angular speed which allows us to obtain the kinetic energy in a form similar to the quantum kinetic energy operator in Eq. 3.4 [Pg.107]

With the development of quantum mechanics at the beginning of the 20th century it became clear that microscopic particles such as atoms, electrons and neutrons, in some cases behave like waves. Both views, the classical picture of a particle characterized by a momentum p, and that of a wave with a wavelength A, are only two different, complementary, viewpoints of the same physical object. Both quantities are related by the de Broglie2 relation A = h/p, where [Pg.162]

Classical thermodynamics ignores microscopic properties such as the behavior of individual atoms and molecules. Nevertheless, a consideration of the classical mechanics of particles will help us to understand the sources of the potential and kinetic energy of a thermodynamic system. [Pg.53]

M. S. Child The comments of Brumer and Quack raise questions about the correspondence between classical and quantum mechanics. In this connection one must first recognize that the classical analogue of a wavepacket is not a single particle but an ensemble. Second, the [Pg.95]

In most textbooks the apparent forces, like the Coriolis and the centrifugal forces, are derived with the help of the framework of classical mechanics of a point particle. [Pg.724]

The main difference between the two models is that, while Bohr considered the electrons to be traditional particles whose motion could be described by the classical mechanics of Newton, the quantum mechanical model treats the electrons as waves. The wave properties of electrons provide a logical explanation for the existence of allowed orbits in Bohr s atomic model. [Pg.470]

In classical mechanics it is proved that an observer who experiments only within a closed system cannot determine whether this system is at rest or is in uniform motion. In fact, the Newtonian equations of motion md xjdt = F (where m is the mass, F the force, X the co-ordinate of a particle, and t the time) remain unchanged if we pass to a moving co-ordinate system by the transformation a == a — vt, provided the force depends only on the position of the particle relative to the co-ordinate system (since [Pg.269]

Wavefunctions describing time-dependent states are solutions to Schrodinger s time-dependent equation. The absolute square of such a wavefunction gives a particle distribution function that depends on time. The time evolution of this particle distribution function is the quantum-mechanical equivalent of the classical concept of a trajectory. It is often convenient to express the time-dependent wave packet as a linear combination of eigenfunctions of the time-independent hamiltonian multiplied by their time-dependent phase factors. [Pg.186]

The variational formalism makes it possible to postulate a relativistic Lagrangian that is Lorentz invariant and reduces to Newtonian mechanics in the classical limit. Introducing a parameter m, the proper mass of a particle, or mass as measured in its own instantaneous rest frame, the Lagrangian for a free particle can be postulated to have the invariant form A = mulxiilx = — mc2. The canonical momentum is pf, = iiiuj, and the Lagrangian equation of motion is [Pg.21]

In conclusion, in this section we have proved that the Markov approximation requires some caution. The Markov approximation may be incompatible with the quantum mechanical nature of the system under study. It leads to the Pauli master equation, and thus it is compatible with the classical picture of a particle randomly jumping from one site to another, a property conflicting, however, with the rigorous quantum mechanical treatment, which yields Anderson localization. [Pg.374]

We shall stress here another aspect in favor of PT. Actually in both non-relativistic and relativistic quantum mechanics one studies the motion (mechanics) of charged particles, that interact according to the laws of electrodynamics. The marriage of non-relativistic mechanics with electrodynamics is problematic, since mechanics is Galilei-invariant, but electrodynamics is Lorentz-invariant. Relativistic theory is consistent insofar as both mechanics and electrodynamics are treated as Lorentz-invariant. A consistent non-relativistic theory should be based on a combination of classical mechanics and the Galilei-invariant limit of electrodynamics as studied in subsection 2.9. [Pg.667]

These wavefunctions are functions that were well known to the people who developed quantum mechanics. They are called spherical harmonics and are labeled Yi, (or Once again, classical mathematics anticipated quantum mechanics in the solution of differential equations. Although the Legendre polynomials do not distinguish between positive and negative values of the quantum number m, the exponential part of the complete wavefunction does. Each set of quantum numbers ( , m ) therefore indicates a unique wavefunction, denoted that can describe the possible state of a particle confined to the surface of a sphere. The wavefunction itself does not depend on either the mass of the particle or the radius of the sphere that defines the system. [Pg.359]

In three dimensions a spatial wave group moves around an harmonic ellipsoid and remains compact, in contrast to the dispersive wave packets of classical optics. The distinction is ascribed to the fact that the quantum wave packet is built up from discrete harmonic components, rather than a continuum of waves. The wave mechanics of a hydrogen electron is conjectured to produce wave packets of the same kind. At small quantum numbers the wave spreads around the nucleus and becomes more particle-like, at high quantum numbers, as it approaches the ionization limit where the electron is ejected from the atom. [Pg.99]

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