Probabilistic interpretation

In our discussion of the electron density in Chapter 5, I mentioned the density functions pi(xi) and p2(xi,X2). I have used the composite space-spin variable X to include both the spatial variables r and the spin variable s. These density functions have a probabilistic interpretation pi(xi)dridii gives the chance of finding an electron in the element dri d i of space and spin, whilst P2(X], X2) dt] d i dt2 di2 gives the chance of finding simultaneously electron 1 in dri dii and electron 2 in dr2di2- The two-electron density function gives information as to how the motion of any pair of electrons is correlated. For independent particles, these probabilities are independent and so we would expect... 

The F lends itself to the usual probabilistic interpretation, that is, F P di is equal to the probability of finding an electron in the volume element dv, thus, a plot of this quantity as a... 

Relationships (61)—(63) admit simple probabilistic interpretation in terms of the branching process. To the reproducing particles of this process the reacted functional groups correspond distinguished by color i and label r. Integer i characterizes the type S, of monomeric unit to which a given group was attached at the moment r of its formation. 

About the same time, Born, who had been lecturing in the United States during 192526, developed a probabilistic interpretation of the motions of the quantized electron. Now Y(x,y,z)2j dt was the probability of finding the electron in a volume element dt at coordinates x,y,z. Born s notion was based on a classical and visual conception of particles, consistent with the positions of atoms established in x-ray crystallogaphy, as discussed in his book, Dynamik der Krystallgitter. 

Before proceeding to discuss the effect of the solvent on the ligand-hgand correlation, we present here a simple probabilistic interpretation of AG. In the canonical ensemble the solvation Helmholtz energy is... 

Constraints on the diagonal element of the density matrix can be useful in the context of the density matrix optimization problem, Eq. (8). As Weinhold and Wilson [23] stressed, the A-representability constraints on the diagonal elements of the density matrix have conceptually appealing probabilistic interpretations this is not true for most of the other known A-representability constraints. 

The second and third inequalities in Eq. (43) are the same, except for a permutation of the indices.) Weinhold and Wilson use the fact that p, represents the probability of observing an electron in orbital i and py represents the probability that orbitals i and j are both occupied to show that each of these constraints has a straightforward probabilistic interpretation. 

The probabilistic interpretation of these conditions is very similar to the interpretation of the (2, 2) conditions. The analysis is readily extended to higher-order densities, where one obtains / + 1 unique constraints on the / -matrix. 

These inequalities are the fourth and fifth Weinhold-Wilson constraints. These (2, 3) inequalities have a probabilistic interpretation very similar to the (2, 2) inequalities. Returning to the fundamental number operators form of the (3, 3) inequalities (cf. Eq. (45)), we see that Eq. (48) was generated from... 

Note that the area (a) also has a more general probabilistic interpretation. 

E. Barbosa and F. Gonzalez, Antiphoton and the Probabilistic Interpretation of Quantum Mechanics, thesis, Univ. National, Bogota, 1995 presented at Colombian Congress of Physics, Bogota, 1999. 

So, it is possible to associate some probabilistic interpretations in the deterministic model. From the probabilistic viewpoint kijAt is the conditional probability that a molecule will be transferred from i to j in the interval t to t +At. Thus k,Al is the conditional probability that a molecule leaves i in that... 

Bridle, J. S. (1990a). Probabilistic interpretation of feedforward classification network outputs, with relationships to statistical pattern recognition. In Neurocomputing Algorithms, Architectures and Applications (ed. F. F. Soulie and J. Herault), pp. 227-36. springer-Verlag, Berlin. 

Born s probabilistic interpretation of was fundamentally different. The probability of atomic events, say the decay of a single neutron, is not a probability of ignorance. Rather, the quantum... 

As explained in Coveil et al. (4) and Cobelli et al. (12), the elements of the mean residence time matrix have important probabilistic interpretations. First, the generic element ijy represents the average time a drug particle entering the system in compartment j spends in compartment i before irreversibly leaving the system by any route. Second, the ratio i equals... 

Upon this bedrock idea, a highly complex and sophisticated statistics has been constructed that can be used to compute not only the frequency of occurrence of most events, but also many kinds of averages such as the expectation values of ensembles of particles. Does this fundamental idea provide a sound basis for all these computations The philosopher Ayer insists that probability theory cannot yield any certainty about future events and that it cannot even indicate what is likely to happen, yet, bearing this in mind, he goes on to state his personal belief that the future will probably resemble the past and so render probabilistic interpretations of our world valid. His presumption would appear to be borne out in practice, especially where large numbers of events or large populations of entities are concerned. 

By analogy, we interpret the square of the wave function i/r for a particle as a probability density for that particle. That is, [i/r(x, y, z)] dV is the probability that the particle will be found in a small volume dY = dxdydz centered at the point (x, y, z). This probabilistic interpretation of the wave function, proposed by the German physicist Max Born, is now generally accepted because it provides a consistent picture of particle motion on a microscopic scale. 

The probabilistic interpretation requires that any function must meet three mathematical conditions before it can be used as a wave function. The next section illustrates how these conditions are extremely helpful in solving the Schrodinger equation. To keep the equations simple, we will state these conditions for systems moving in only one dimension. All the conditions extend immediately to three dimensions when proper coordinates and notation are used. (You should read Appendix A6, which reviews probability concepts and language, before proceeding further with this chapter.)... 

The probabilistic interpretation of (G.3) and (G.6) is also straightforward. Placing a particle at in an ideal gas system of exactly N particles, changes the conditional probability (or density) of finding a particle at any location in the system from N/Vinto (N—1)1 V hence, the correlation function has the form (G.3). In an open system, placing a particle at a fixed position does not affect the density at any other point in the system. This is so since the chemical potential p, rather than N is fixed, hence the density at any point in the system is constant, (N)/V, figure G.l. 

We next turn to the probabilistic interpretation of (G.19). We have seen that for an ideal gas there is a very simple probabilistic interpretation of behavior (G.3), and we have seen that the N 1 term is a result of the closure correlation. 

The probabilistic interpretation of gc(CC) in (G.19) is not as obvious, and it is a little more tricky than in the case of an ideal gas. We provide here the appropriate probabilistic interpretation of the closure correlation in (G.19). 

In the case of the one-component system, we have noted that the probabilistic interpretation of (G.19) is not so obvious as in the case of an ideal gas. It is less obvious for the case of mixtures. In order to interpret (G.30) probabilistically, we proceed to do a similar transformation of equation (G.30). From (G.28) and (G.30), we obtain the general result... 

The way contextual correspondences are defined, they are deterministic, not allowing a probabilistic interpretation of a correspondence. Therefore, contextual correspondences are meant to resolve an issue of uncertainty by finding a more refined, yet deterministic, correspondence. It is worth noting, however, that an introduction of stochastic analysis already exists in the form of statistical significance, which can be extended to handle probabilistic mappings as well. 

Quantum distributions, denoted p (p,q), do not have a probabilistic interpretation. Rather, the conditions closest to those in Eq. (3.31) are... 

Equation (3.32a) implies normalization, and Eq. (3.32b) contains the essential probabilistic interpretation of the projections onto the momenta or coordinates. The last condition, a natural consequence of the definitions of the density matrix and the Wigner-Weyl transform, explicitly eliminates the singular distributions allowed in Eq. (3.31d). That is, although the completeness of the quantum PijXp, q) basis permits the construction of 8 function distributions, they make, unlike classical mechanics, no natural appearance in quantum mechanics wherein eigenfunctions of LQ are square integrable and such singular distributions are explicitly excluded in Eq. (3.32d). 

---