Weighted superpositions

This ensemble is a weighted superposition of NVT ensembles with different values, of N. As a rule of tiuiinb, a typical MC sweep consists of N attempted moves, each of which is chosen randomly to be (i) a displacement (liandled exactly as in constant-AfTMC) (ii) the creation of a new particle at a randomly selected position (iii) the destruction of a randomly selected particle from the system. The probabilities for attempting creation and destniction must be equal (for consistency with what follows), but they need not be equal to the probability for attempting displacement (although they often are). 

When a mixture containing compounds with known spectra is analysed, it is possible to determine the mixture s composition. According to the additivity law (equation (11.10)), the spectrum of the mixture corresponds to the weighted superposition of each of the individual constituents. The following section recalls the classical calculation method. It is rarely used as such because it is typically incorporated into software. 

VB (RVB) ansatze, where the trial wave-functions are a weighted superposition of singlets, each singlet being a product of N two-body singlets on two (not necessarily nearest-neighbor) sites at a time. 

Within the RVB scheme for the ground state, the simplest wave function beyond that of a single Kekule structure can be defined as a simple equally-weighted superposition of Kekule structures (EWK) within a given LR-SPO subspace,... 

One-bond-range RVB ansatze A one-bond-range RVB ansatz o showing LR-SPO D is a weighted superposition of dimer-covering configurations with LR-SPO D. They can be written as... 

The SCM concept combined with a powerful sorption database allows a straightforward extension towards rocks and soils composed of several minerals. Their sorption behaviour should be predicted from the weighted superposition of the sorption processes for all constituting minerals. The number of important rock-forming minerals is rather limited, more then 95 % of the known rocks in the earth s crust consist of only about 20 minerals and mineral groups (Ronov and Yaroshevsky, 1967). Thus we hypothesized that the sorption behaviour of uranium(VI) onto rocks can be described... 

Figure 8.3 solid-state NMR of [C2Py][NTfj]-t/,Q/silica (40 wt% IL loading on silica) while heating, (a) Experimental spectra, (b) Simulated spectra as the weighted superposition of the spectra at low (215 K) and high (274 K) temperatures [10]. 

Fig. 3. Ljjj absorption spectra of Sm in trivalent Sm Oj (lower spectrum) and in mixed valent SmB (upper spectrum). The absorption maxima in both compounds are observed at identical ener es. SmB, however, exhibits a significant shoulder at about — 8eV displaced from the maximum of the trivalent absorption line (adapted from Vainshtein et aL (1965a)). The second absorption line indicates the presence of divalent Sm in SmB ( = 2.65 0.05). The valence number v is obtained from a weighted superposition d two single-shaped Lm absorption spectra disjdaced by 8eV (Vainshtein et al-1965b).

In the theory of free oscillations of the earth it is assumed that the earth is a spherically symmetric elastic body. Takeuchi and Saito obtained the normal modes for the radially heterogeneous earth. Two Independent modes are identified toroidal modes, corresponding to a twisting motion of the sphere (SM waves), and spheroidal modes, corresponding to a distorting motion of the sphere (P-SV waves). Once the normal modes are obtained, the motions of the earth can be expressed by the weighted superposition of all normal modes. 

Similarly, the bivariate molecular weight and chemical composition distributions of binary copolymers made with multiple-site catalysts can be described as a weighted superposition of Stockmayer s distributions [39]. If we consider only the chemical composition component of the distribution, as described by Eq. (22), the distribution of polymer made with a multiple-site catalyst becomes Eq. (28). 

The band at 841/cm was chosen for the crystalline phase and the band at 973/cm for the amorphous phase. Because the spectra are the weighted superposition of single absorption peaks, all the absorbances were obtained by fitting the experimental spectra with a weighted combination of single peaks, adopting Gaussian/Lorentzian peak functions. 

Assume that at time f = 0, a harmonic oscillator has been prepared via some prior time-dependent perturbation such that its time-dependent wavefunction at that instant is an equally weighted superposition of fhe n = 0 and n = l time-independent wavefunctions. Write the explicit form of P(x, t) and derive an expression for the probability density at the point x = 0 as a function of time for f > 0. 

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