Motion translational

J-C is called the Hamiltonian operator. It describes the forces relevant for the problem of interest. In classical mechanics, the sum of kinetic plus potential energies is an invariant conserved quantity, the total energy (see Equation (.3.6)). In quantum mechanics, the role of this invariant is played by the Hamiltonian operator. For example, for the one-dimensional translational motion of a particle having mass m and momentum p, the Hamiltonian operator is 

While classical mechanics regards p and V as functions of time and spatial position, quantum mechanics regards p and y as mathematical operators that create the right differential equation for the problem at hand. For example, the translational momentum operator is p = ( 14n ) d Idx ) in one 

The Particle-in-a-Box Is the Quantum Mechanical Model for Translational Motion 

The particle-in-a-box is a model for the freedom of a particle to move within a confined space. It applies to electrons contained within atoms and molecules, and molecules contained within test tubes. Let s first solve a one-dimensional problem. 

Equation (11.4) then becomes a linear second-order differential equation, 

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The following derivation is modified from that of Fowler and Guggenheim [10,11]. The adsorbed molecules are considered to differ from gaseous ones in that their potential energy and local partition function (see Section XVI-4A) have been modified and that, instead of possessing normal translational motion, they are confined to localized sites without any interactions between adjacent molecules but with an adsorption energy Q. 

The two expressions for bo may be brought into formal identity as follows. On adsorption, the three degrees of translational freedom can be supposed to appear as two degrees of translational motion within the confines of a two-... 

The use of lasers to cool atomic translational motion has been one of the most exciting developments in atomic physics in the last 15 years. For excellent reviews, see [66, 67]. Here we give a non-orthodox presentation, based on [68]. 

In an ideal molecular gas, each molecule typically has translational, rotational and vibrational degrees of freedom. The example of one free particle in a box is appropriate for the translational motion. The next example of oscillators can be used for the vibrational motion of molecules. 

To illustrate, consider an ideal classical gas of Nmolecules occupying a volume Vand each with mass Maud tliree degrees of translational motion. The Flamiltonian is... 

These results do not agree with experimental results. At room temperature, while the translational motion of diatomic molecules may be treated classically, the rotation and vibration have quantum attributes. In addition, quantum mechanically one should also consider the electronic degrees of freedom. However, typical electronic excitation energies are very large compared to k T (they are of the order of a few electronvolts, and 1 eV corresponds to 10 000 K). Such internal degrees of freedom are considered frozen, and an electronic cloud in a diatomic molecule is assumed to be in its ground state f with degeneracy g. The two nuclei A and... 

An important further consequence of curvature of the interaction region and a late barrier is tliat molecules that fail to dissociate can return to the gas-phase in vibrational states different from the initial, as has been observed experunentally in the H2/CU system [53, ]. To undergo vibrational (de-)excitation, the molecules must round the elbow part way, but fail to go over the barrier, eitlier because it is too high, or because the combination of vibrational and translational motions is such that the molecule moves across rather than over the barrier. Such vibrational excitation and de-excitation constrains the PES in that we require the elbow to have high curvature. Dissociation is not necessary, however, for as we have pointed out, vibrational excitation is observed in the scattering of NO from Ag(l 11) [55]. 

The treatment of translational motion in three dimensions involves representation of particle motions in tenns... 

All the theory developed up to this point has been limited in the sense that translational motion (the continuum degree of freedom) has been restricted to one dimension. In this section we discuss the generalization of this to three dimensions for collision processes where space is isotropic (i.e., collisions in homogeneous phases, such as in a... 

The biggest change associated with going from one to tliree dimensional translational motion refers to asymptotic boundary conditions. In tiiree dimensions, the initial scattering wavefiinction for a single particle... 

Kroes G J, Wiesenekker G, Baerends E J, Mowrey R C and Neuhauser D 1996 Dissociative chemisorption of H2 on Cu(IOO)—a four-dimensional study of the effect of parallel translational motion on the reaction dynamics J. Chem. Phys. 105 5979... 

As stated earlier, within C(t) there is also an equilibrium average over translational motion of the molecules. For a gas-phase sample undergoing random collisions and at thermal equilibrium, this average is characterized by the well known Maxwell-Boltzmann velocity distribution ... 

This result, when substituted into the expressions for C(t), yields expressions identieal to those given for the three eases treated above with one modifieation. The translational motion average need no longer be eonsidered in eaeh C(t) instead, the earlier expressions for C(t) must eaeh be multiplied by a faetor exp(- co2t2kT/(2me2)) that embodies the translationally averaged Doppler shift. The speetral line shape funetion I(co) ean then be obtained for eaeh C(t) by simply Fourier transforming ... 

The reaction path shows how Xe and Clj react with electrons initially to form Xe cations. These react with Clj or Cl- to give electronically excited-state molecules XeCl, which emit light to return to ground-state XeCI. The latter are not stable and immediately dissociate to give xenon and chlorine. In such gas lasers, translational motion of the excited-state XeCl gives rise to some Doppler shifting in the laser light, so the emission line is not as sharp as it is in solid-state lasers. 

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Center-of-mass translational motion in MD simulations is often quantified in tenns of diffusion constants, D, computed from the Einstein relation. 

Rotational flow occurs in an element of a fluid that rotates about its axis, in addition to having translational motion (e.g., water passing through a paddle wheel). 

Potential energy surfaces are also central to our quantum-mechanical studies, and we are going to meet them again and again in subsequent chapters. Let s start then with Figure 3.1, which shows H2+. We are not going to be concerned with the overall translational motion of the molecule. For simphcity, I have drawn a local axis system with the centre of mass as the origin. By convention, we label the intemuclear axis the z-axis. 

In a liquid that is in thermodynamic equilibrium and which contains only one chemical species, the particles are in translational motion due to thermal agitation. The term for this motion, which can be characterized as a random walk of the particles, is self-diffusion. It can be quantified by observing the molecular displacements of the single particles. The self-diffusion coefficient is introduced by the Einstein relationship... 

Fig. 7-8. Types of motion of a molecule of carbon dioxide, C(>2. A. Translational motion the molecule moves from place to place. B. Rotational motion the molecule rotates about its center of mass. C. Vibrational motion the atoms move alternately toward and away from the center of mass.

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