Dimensioning limit

Kais, S, Sung, S.M., Herschbach, D.R., Large-Z and -N Dependence of Atomic Energies of the Large-Dimension Limit, International Journal of Quantum Chemistry, 1994 49 657-674. 

Dispersive transport in PVC was investigated. The results of Pfister and Griffits obtained by the transit method are shown in Fig. 6. The hole current forms at temperatures > 400 K clearly show a bend corresponding to the transit time of the holes. At lower temperature the bend is not seen and transit time definition needs special methods. The pulse form shows the broad expansion during transition to the opposite electrodes. This expansion corresponds to the dispersive transport [15]. The super-linear dependence of the transit time versus sample thickness did not hold for pure PVC. This is in disagreement with the Scher-Montroll model. There are a lot of reasons for the discrepancy. One reason may be the influence of the system dimensions. It is quite possible that polymer chains define dimension limits on charge carrier transfer. 

One kind of a multicritical point is a point over a critical line where more than two different states coalesce. The common multicritical points in statistical mechanics theory of phase transition are tricritical points (the point that separates a first order and a continuous line) or bicritical points (two continuous lines merge in a first order line) (see, for example, Ref. 166). These multicritical points were observed in quantum few-body systems only in the large dimension limit approximation for small molecules [10,32]. For three-dimensional systems, this kind of multicritical points was not reported yet. 

Dimensions Limited by log diameter Wider products possible... 

It is clearly seen that, for a given (, the scalar products become closer to unity with increasing space dimension. We conclude that the generalized state (78) approaches the conventional squeezed vacuum in the dimension limit. 

By comparing Eq. (95) with Eqs. (88) and (94), we can conclude that the differences between the generalized and truncated squeezed vacuums are smaller than those between them and the conventional squeezed vacuum. All these states coincide in the high-dimension limit. In Fig. lOb-f, we have presented the Wigner representation of the generalized squeezed vacuum for those values of the squeeze parameter , which correspond to maximum and minimum values of the vacuum-state probability given by Ps(0) = 2 (see Fig. 10a). We... 

The problem reduces to derivation of the coefficients d satisfying the condition in the dimension limit... 

While small (ca. <100nm), particulate samples can be viewed direcdy after shadowcasting (pseudoreplicas), larger samples require that the biological material be stripped away to form a high resolution replica. Replica techniques are advantageous over other methods, here, in that thicker specimens can be used. With this approach, only the evaporator device dimensions limit primary sample size. 

The USP uses a filtration technique whereby the particles are retained by a membrane filter and then measured under lOOxmagnification. Particles are gauged by their effective linear dimension. Limits are ... 

Mb > 3-1000 TeV (astrophys. and cosmology 2-flat dimensions limits depend on technique and assumptions)... 

Figure 1. Evolution of the radial electron density for argon from each of the three limits described in Table 1. The limits (top row) are those given by the Thomas-Fermi approximation, the non-interacting electron model, and the large-dimension limit. The abscissae are Hn-ear in y/r, where r is given in units of (IS/JV) for the first column, (18/Z) Co for the second, and (D/Zy a<, for the third. All curves except the Thomas-Fermi limit were obtained by adding Slater distributions whose exponents were determined from the subhamiltonian minima described in Sec. 4.

Eqs. (5) and (7) can be thought of as changes in the distance and energy units, respectively. In these new, dimension-scaled, units the radial expectation value, (r) = has a much tamer dimension dependence, and the new groimd-state energy, E = — is completely independent of D. Thus, the one-dimension limit of the scaled Schrodinger equation, Eq. (6), is not such a bad model for the physical, three-dimensional, problem. 

The second and third terms on the right-hand side are finite at 2 = 1, so they become insignificant compared to the first term in the one-dimension limit. Note that 8 r) = 2S(x), since the former is normalized on the domain 0 < r < oo while the latter is normalized on a domain that is twice as large, —oo < x < oo. Thus,... 

Hence, our prescription for constructing the one-dimension limit of a Schiddinger equation is to replace the coulomb potential 1/r with (r). 

Thus, the one-dimension limit for the polarizability does not give the full solution however, it differs only by a quadratic polynomial in D. 

D.D. Frantz and D.R. Herschbach, Lewis electronic structures as the large- dimension limit for and H2 molecules, Chem. Phys. 126 (1988) 59-71. 

The large-dimension limit offers an approach that explicitly includes electron correlation [2,3]. It seems to give an excellent qualitative description of correlation effects that are difficult to understand in terms of independent-electron models [3,4]. This is because all the coulombic forces are retained in their exact form. The dimensionality of space is treated as an arbitrary and continuous parameter, D. For any atom or molecule, an exact solution can be obtained in the limit D oo. The problem then becomes how to exploit this limiting result in constructing an approximate solution for the actual physical system with D = 3. Our aim is twofold to find an efficient way to calculate accurate expectation values, and to develop reliable qualitative interpretations of the D oo results that elucidate the correlation effects at i = 3. 

The most straightforward way to evaluate corrections to the large-dimension limit is to compute a 1/D expansion [5]. The energy eigenvalues each have a unique asymptotic expansion in the form... 

Eq. (2) does not provide a useful large dimension limit, since the centrifugal term becomes infinite. 

The one-dimension limit of Eq. (2) as written is also not well-defined. The problem now is that the expectation value (V), and hence the energy, becomes infinite at particle coalescences [18], as we discussed in Chapter 4.1. V) can be rendered finite through... 

The large-dimension limit is formally equivalent to a system of classical particles each confined by a potential well [13]. The effective masses of the particles are proportional to ). For 8 small but nonzero the particles oscillate about the minima of their wells, and in the limit 0 they all come to rest. The scaled energy eigenvalue at 5 = 0 is... 

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