The Physical Meaning

Based on elementary number theory, we have arrived at a computational scheme that works unexpectedly well for the simulation of chemical phenomena. The problem is to find a plausible interpretation to connect the abstract model with the concrete. The statistical scheme of probability densities and point particles adopted in physics does not provide a satisfactory answer in this case. We are forced to enquire more deeply into the fundamental nature of matter. 

The only fundamental theory that considers the genesis of matter is the theory of general relativity. It is formulated in 4D space-time as a set of field equations, 

For convenience, humans consider their living space to be Euclidean, and even the most advanced cosmologies are still formulated in such terms. In order to operate in Euclidean space, it is necessary to separate the mathematically equivalent variables of Einstein s equation into a universal time variable and the three familiar variables of coordinate space. This operation destroys the 4D field equations and no longer provides any insight into the nature of matter. For this reason, the existence of matter is added to physical theory as an ad hoc postulate—the prescription of classical Newtonian mechanics. 

Quantum mechanics follows the same prescription, by separating the 4D potential function, formulated as 

The suspicion that this separation of variables, although mathematically sound, leads to a less than perfect description of quantum systems is confirmed [24] by statistical testing of the seminal equation that relates the frequency of the energy radiated or absorbed by a H atom to the integers in the Rydberg formula 

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It turns out to be considerably easier to obtain fairly precise measurements of a change in the surface free energy of a solid than it is to get an absolute experimental value. The procedures and methods may now be clear-cut, and the calculation has a thermodynamic basis, but there remain some questions about the physical meaning of the change. This point is discussed further in the following material and in Section X-6. 

The physical meaning of the sections on transformation matrices and unitary matrices is that we can try to rotate our coordinate system so that each component... 

It should be noted that the Hartree-Fock equations F ( )i = 8i ([)] possess solutions for the spin-orbitals which appear in F (the so-called occupied spin-orbitals) as well as for orbitals which are not occupied in F (the so-called virtual spin-orbitals). In fact, the F operator is hermitian, so it possesses a complete set of orthonormal eigenfunctions only those which appear in F appear in the coulomb and exchange potentials of the Foek operator. The physical meaning of the occupied and virtual orbitals will be clarified later in this Chapter (Section VITA)... 

Similarity Variables The physical meaning of the term similarity relates to internal similitude, or self-similitude. Thus, similar solutions in boundaiy-layer flow over a horizontal flat plate are those for which the horizontal component of velocity u has the property that two velocity profiles located at different coordinates x differ only by a scale factor. The mathematical interpretation of the term similarity is a transformation of variables carried out so that a reduction in the number of independent variables is achieved. There are essentially two methods for finding similarity variables, separation of variables (not the classical concept) and the use of continuous transformation groups. The basic theoiy is available in Ames (see the references). 

Wherein t is the thickness and is the distance to the centroid of the k orthotropic layer. What is the physical meaning of the coefficients of in each of the foregoing expressions . ... 

According to Eq. (4-62), when woTo < 1, T, is proportional to 1/Tc, whereas when woTc 1, Ti is proportional to Tc. When Tc = Wo, Tj has its minimum value. Figure 4-7 is a schematic representation of the relationship between T and Tc. The physical meaning of this relationship is that coupling between the spin system and the lattice is most efficient when the resonance frequency and the frequency of molecular motion are equal. Tc can be measured by studying the dependence of Ti on wq (by varying the field strength). For small molecules in solution Tc is commonly 10 to 10 s. 

In many electron atoms the maximum contributions to the polarizability and to London forces arise from configurations with more than one electron contributing to the net dipole moment of the atom. But in such configurations the electronic repulsion is especially high. The physical meaning to be attributed to the Qkl terms is just the additional electron repulsive energy which these configurations require. 

For cpr < 1 Mooney formula transforms in a natural way into Einstein equation, and for cpr -> 1 this formula predicts thatp -> oo, which corresponds to the physical meaning of the phenomenon. 

This brief discussion of the physical meaning and mutual correspondence of different models and theories of rotational motion is intended as a guide for those who do not intend to examine the book systematically. Setting forth the material consistently, one cannot avoid certain formalisms peculiar to angular momentum theories. We hope, however, that a detailed commentary will enable readers to form a clear notion of the most important assumptions and results without referring to proofs. 

The physical meaning of and f L.., is obvious they govern the relaxation of rotational energy and angular momentum, respectively. The former is also an operator of the spectral exchange between the components of the isotropic Raman Q-branch. So, equality (7.94a) holds, as the probability conservation law. In contrast, the second one, Eq. (7.94b), is wrong, because, after substitution into the definition of the angular momentum correlation time... 

Returning to the Figure 5.2a we can further discuss the physical meaning of A to gain some more physical insight and to prove the validity of the sacrificial promoter concept. 

The physical meaning of the parameter 2FNG/I is obvious It expresses the time required to form a monolayer of oxide ions on a surface with NG adsorption sites when the oxide ions are supplied at a rate I/2F. This proves that NEMCA is a surface phenomenon (not a bulk phenomenon and not a phenomenon at the tpb) taking place over the entire gas-exposed catalyst electrode surface. 

In section 5.4.3 we have discussed the physical meaning and range of validity of the potential-work function equivalence equations of solid state electrochemistry ... 

Although we do not wish to imply that equation (6.20) is a general fundamental equation, we are also not aware of any published exceptions to the physical meaning it conveys, i.e. that the enthalpy of adsorption and thus, according to any isotherm, the coverage of an electron acceptor/donor adsorbate decreases/ increases with increasing work function O and thus decreasing Fermi level EF. 

The dependence of P (PeL) and g (PeL) is shown in Fig. 11.4. The parameter P (PeL) is a parabola with an axis of symmetry left of the line Pcl = 0. Since the Peclet number is positive, for any value of the operating parameters, the physical meaning is that only for the right branch of this parabola, which intersects the axis of the abscissa at some critical value of Peclet number, Pcl = Peer- The vertical line PeL = Peer subdivides the parametrical plane P - Pcl into two domains, corresponding to positive (PeL < Peer) or negative (PeL > Peer) values of the parameter P . The critical Peclet number is... 

The curve ( (PeL) is a cubic parabola, which passes through the point 0(0,0). Since, the Peclet number is positive, the physical meaning has the falling and rising branches of ( (PeL), which are located on the right part of the parameter plane < -PeL. 

The physical meaning of the electron propagator rests chiefiy in its poles (energies where singularities lie) and residues (coefficients of the terms responsible for the singularities) [1]. In its spectral form, the r,s element of the electron... 

This equation describes the series of lines in Figure 5, the variable parameter being represented by The physical meaning of coefficients aj follows from comparison of eqs.(17), (18) and (19) ao equals logAo and ai=-Eo/ 2.303 RT, the subscripts 0 referring to the standard substituent, az =p ,at the infinite temperature, and as = -/3pco. Hence, 0 is obtained as -aj/a2. Direct correlations of AH and AS with a (176, 197, 198) or other parameters (199, 200) are usually bad and cannot serve to obtain the AH/AS relationship. 

The physical meaning of the constant (3, connected with the reversal of reactivity at the temperature T = /3, is a puzzling corollary of the isokinetic relationship, noted already by older authors (26, 28) and discussed many times since (1-6, 148, 149, 151, 153, 163, 188, 212). Especially when the relative reactivity in a given series is explained in theoretically significant terms, it is hard to believe that the interpretation could lose its validity, when only temperature is changed. The question thus becomes important of whether the isokinetic temperature may in principle be experimentally accessible, or whether it is merely an extrapolation without any immediate physical meaning. 

To conclude this section, we can state that all of the theories presented hitherto, even when starting from general principles, inevitably embody several assumptions, which in fact represent the heart of the analysis. However, the physical meaning of these assumptions usually is not known, so that no theory is able to predict in which reaction series isokinetic behavior appears and in which it does not. Neither is the structural theory of organic chemistry able to make such a prediction and to define the terms reaction series or similar reactions or small structure changes it can only afford many examples. 

The physical meaning of B = mn is that no bifurcation into the diabatic state 3 > occurs at X2 on the second half of the sweep whatever the probability p2 is. The conditions of pi and C guarantee that the interference between 1 > and 2 > at Xi on the way back leads to the complete excitation to 2 >. The complete excitation to 3 > can be achieved by one period of sweeping, if we start from 0)2 The condition is given by... 

Figure 8. The physical meaning of the absolute electrode potential.

The physical meaning of the g (ion) potential depends on the accepted model of an ionic double layer. The proposed models correspond to the Gouy-Chapman diffuse layer, with or without allowance for the Stem modification and/or the penetration of small counter-ions above the plane of the ionic heads of the adsorbed large ions. " The experimental data obtained for the adsorption of dodecyl trimethylammonium bromide and sodium dodecyl sulfate strongly support the Haydon and Taylor mode According to this model, there is a considerable space between the ionic heads and the surface boundary between, for instance, water and heptane. The presence in this space of small inorganic ions forms an additional diffuse layer that partly compensates for the diffuse layer potential between the ionic heads and the bulk solution. Thus, the Eq. (31) may be considered as a linear combination of two linear functions, one of which [A% - g (dip)] crosses the zero point of the coordinates (A% and 1/A are equal to zero), and the other has an intercept on the potential axis. This, of course, implies that the orientation of the apparent dipole moments of the long-chain ions is independent of A. 

We think that the physical meaning of the SRH matrices is not yet fully understood and can therefore be considered an open field of research. It is to be expected that a more complete understanding of the SRH matrices will lead to new applications of this formalism. 

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