One-particle

The one-dimensional cases discussed above illustrate many of die qualitative features of quantum mechanics, and their relative simplicity makes them quite easy to study. Motion in more than one dimension and (especially) that of more than one particle is considerably more complicated, but many of the general features of these systems can be understood from simple considerations. Wliile one relatively connnon feature of multidimensional problems in quantum mechanics is degeneracy, it turns out that the ground state must be non-degenerate. To prove this, simply assume the opposite to be true, i.e. 

Applications of quantum mechanics to chemistry invariably deal with systems (atoms and molecules) that contain more than one particle. Apart from the hydrogen atom, the stationary-state energies caimot be calculated exactly, and compromises must be made in order to estimate them. Perhaps the most useful and widely used approximation in chemistry is the independent-particle approximation, which can take several fomis. Conuiion to all of these is the assumption that the Hamiltonian operator for a system consisting of n particles is approximated by tlie sum... 

For an isotropic fluid, the one-particle correlation fimction is independent of tire position and... 

We conclude this section by discussing an expression for the excess chemical potential in temrs of the pair correlation fimction and a parameter X, which couples the interactions of one particle with the rest. The idea of a coupling parameter was mtrodiiced by Onsager [20] and Kirkwood [Hj. The choice of X depends on the system considered. In an electrolyte solution it could be the charge, but in general it is some variable that characterizes the pair potential. The potential energy of the system... 

No more than one particle may occupy a cell, and only nearest-neighbour cells that are both occupied mteract with energy -c. Otherwise the energy of interactions between cells is zero. The total energy for a given set of occupation numbers ] = (n, of the cells is then... 

The projector augmented-wave (PAW) DFT method was invented by Blochl to generalize both the pseudopotential and the LAPW DFT teclmiques [M]- PAW, however, provides all-electron one-particle wavefiinctions not accessible with the pseudopotential approach. The central idea of the PAW is to express the all-electron quantities in tenns of a pseudo-wavefiinction (easily expanded in plane waves) tenn that describes mterstitial contributions well, and one-centre corrections expanded in tenns of atom-centred fiinctions, that allow for the recovery of the all-electron quantities. The LAPW method is a special case of the PAW method and the pseudopotential fonnalism is obtained by an approximation. Comparisons of the PAW method to other all-electron methods show an accuracy similar to the FLAPW results and an efficiency comparable to plane wave pseudopotential calculations [, ]. PAW is also fonnulated to carry out DFT dynamics, where the forces on nuclei and wavefiinctions are calculated from the PAW wavefiinctions. (Another all-electron DFT molecular dynamics teclmique using a mixed-basis approach is applied in [84].)... 

Hedin L 1965 New method for calculating the one-particle Green s function with application to the electron-gas problem Pbys. Rev. 139 A796... 

The total wavefunction r2,. . ., r is written as a product of single-particle functions (Hartree approximation). The various integrals are evaluated in tire saddle point approximation. A simple Gaussian fomr for tire trial one-particle wavefunction... 

The main difference between the force-bias and the smart Monte Carlo methods is that the latter does not impose any limit on the displacement that m atom may undergo. The displacement in the force-bias method is limited to a cube of the appropriate size centred on the atom. However, in practice the two methods are very similar and there is often little to choose between them. In suitable cases they can be much more efficient at covering phase space and are better able to avoid bottlenecks in phase space than the conventional Metropolis Monte Carlo algorithm. The methods significantly enhance the acceptance rate of trial moves, thereby enabling Icirger moves to be made as well as simultaneous moves of more than one particle. However, the need to calculate the forces makes the methods much more elaborate, and comparable in complexity to molecular dynamics. 

The range of integration and the meaning of the variables x will be defined by the specific problem of interest for example, in polar coordinates for one particle, x r,0,( ), and for N... 

Unimolecular (Section 4 8) Describing a step in a reaction mechanism in which only one particle undergoes a chemi cal change at the transition state... 

Skin Flotation. Hydrophobic particles can be removed in the form of a thin, usually one particle thick layer on top of a trough, giving rise to the skin flotation process. 

In this decay process, only one particle is emitted and, because energy is conserved, for each level in the daughter nucleus there is a unique a-particle energy. This means that a measurement of the differences in the energies of the a-particles emitted in a radioactive decay gives expHcidy the differences in the energies of the levels in the daughter nucleus. 

In this process only one particle is emitted, so the energy spectmm of the neutrinos consists of discrete lines and in principle the energies of the levels in the daughter nucleus could be deterrnined from this spectmm. However, the detection of neutrinos is very difficult, so this is not a practical possibihty. 

The well-known DLVO theory of coUoid stabiUty (10) attributes the state of flocculation to the balance between the van der Waals attractive forces and the repulsive electric double-layer forces at the Hquid—soHd interface. The potential at the double layer, called the zeta potential, is measured indirectly by electrophoretic mobiUty or streaming potential. The bridging flocculation by which polymer molecules are adsorbed on more than one particle results from charge effects, van der Waals forces, or hydrogen bonding (see Colloids). 

Phase Doppler particle analyzers are essentially single-particle counters because they measure one particle at a time within a small sampling volume. This volume must be kept small to minimize the probabiUty of having more than one droplet in the volume at any given instant. This probabiUty increases as the concentration of droplets becomes greater, and there is more risk of measurement errors. 

The word particle has become so widely used ia the technical mbber and carbon black Hterature that it is convenient to retain the term when ia fact nodule is meant. The layer planes are curved, distorted, and of varyiag size. They also iatersect and interconnect one particle or nodule with its neighbors. This type of stmcture has been termed paracrystalline. It is obvious that iadividual particles do not exist ia carbon blacks, with the exception of thermal... 

Stream scanning Brinkmann, Glimet, Goulter, Dantec, Erdco, Faley, Flowvision, Hiac/Royco, Kowa, Lasentec, Malvern, Met One, Particle Measuring Systems, Polytec, Procedyne, Rion, Spectrex 0.2-10,000 lm O.l-lOg (also on-line)... 

If 0 fluid streomline passes within one particle rodius of the collecting body, o particle traveling olong the streamline will touch the body and may be collected without the influence of inertia or brownian diffusion. 

Note that this is also a functional of liaAr), Cas(r), and 4 ). Imposing constraints concerning the orthonormality of the configuration state function (C) and one-particle orbitals (pi) on the equation, one can derive the Eock operator from. A based on the variational principle ... 

Equation (38) gives the probability that one particle has fractured. However there are n=b/a particles in the entire row, therefore ... 

It is probable that, in practice, (j) will lie between 0.5 and 1.0 but, to simplify the argument, (j) will be taken as unity. Thus, it will be assumed that one lateral step will be taken by a given molecule for every step traveled axially that is equivalent to one particle diameter. 

Attrition The rubbing of one particle against another in a resin bed frictional wear that will affect the site of resin particles. 

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