Physical interpretation

The physical interpretation of the periodic relationship suggested by numerical pattern generation relies on the known effect of isotropic compression on the electronic structure of atoms [24]. Compression causes all energy levels to rise and removes the degeneracy of sub-levels. The effect becomes more pronounced with increasing quantum number l. Relative energies for hydro- 

This sequence approaches the inversion inferred above. Hence the degrees of inversion represented by the spectra indicated at ratios 1.0 and 1.04, require enormous pressures that can be generated only in massive stars. To understand what happens in such a star we note that a pair of neighbouring points on any festoon mapped in figure 2 differs by the equivalent of an a-particle, with a protonmeutron ratio unity. A logical picture that relates to nuclear synthesis emerges. 

Pressure within stars can increase to the point where sub-atomic particles fuse to form 2H, 3He, 4He, and 5He. The a-particle is the most stable of these units and becomes formed in sufficient excess to add progressively to each of four starting units to produce nuclides in four series of mass number An, 4n 1, An — 2, as observed [21]. Under these conditions the protonmeutron ratio for each series approaches unity with increasing mass number. At a certain age, a star of such magnitude explodes as a supernova to release the synthesized material into low-pressure environments in which a phase transition ensues. This transition consists of an inversion of energy levels 

Suppose we have a dynamical system whose state IT) can be calculated by solving the Schrodinger equation with the appropriate boundary conditions. An experiment is set up to observe a particular dynamical property of the system corresponding to an observable whose eigenstates are 0). 

The experiment observes an ensemble of events, each with the same initial conditions (as nearly as can be physically achieved). The number n of observations of the system when it is in a particular state 0) is recorded. The number n, suitably normalised, is taken as an estimate of the probability of finding the system in the state 0). The standard error of the estimate is 

The physical interpretation of quantum mechanics is as follows. The probability amplitude / of finding the system in the state 0) is 

Bound states are normalisable. If we represent all the coordinates of the system by x then the coordinate representation of N is 

The integrand is the probability of finding the system with coordinates x. If the integrand tends rapidly enough to zero when all parts of the system are remote from the centre of mass then the integral is convergent. I T) is defined by choosing iV=l. The total probability is 1. 

Electromagnetic wave propagating along a chain of oppositely charged ions can excite a transverse optical (TO) mode. To excite a longitudinal optical (LO) mode, the k-vector must make an angle with the chain in order to project a component of its E-vector along the chain (Berreman effect). 

It should be noted that sp -bonded materials such as Si and Ge also exhibit both an optical and an acoustic branch even though they contain only one type of atom. The covalent sp bond places much of the electron density between the atoms so that the charge distribution is similar to that of an ionic system. 

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Like the geometry of Euclid and the mechanics of Newton, quantum mechanics is an axiomatic subject. By making several assertions, or postulates, about the mathematical properties of and physical interpretation associated with solutions to the Scluodinger equation, the subject of quantum mechanics can be applied to understand behaviour in atomic and molecular systems. The fust of these postulates is ... 

We now proceed to some examples of this Fourier transfonn view of optical spectroscopy. Consider, for example, the UV absorption spectnun of CO2, shown in figure Al.6.11. The spectnuu is seen to have a long progression of vibrational features, each with fairly unifonu shape and width. Wliat is the physical interpretation of tliis vibrational progression and what is the origin of the width of the features The goal is to come up with a dynamical model that leads to a wavepacket autocorrelation fiinction whose Fourier transfonn... 

In words, equation (Al.6.89) is saying that the second-order wavefunction is obtained by propagating the initial wavefunction on the ground-state surface until time t", at which time it is excited up to the excited state, upon which it evolves until it is returned to the ground state at time t, where it propagates until time t. NRT stands for non-resonant tenn it is obtained by and cOj -f-> -cOg, and its physical interpretation is... 

The pair correlation fiinction has a simple physical interpretation as the potential of mean force between two particles separated by a distance r... 

The physical interpretation of the scattering matrix elements is best understood in tenns of its square modulus... 

Bain A D and Duns G J 1996 A unified approach to dynamic NMR based on a physical interpretation of the transition probability Can. J. Chem. 74 819-24... 

In the present study we try to obtain the isotherm equation in the form of a sum of the three terms Langmuir s, Henry s and multilayer adsorption, because it is the most convenient and is easily physically interpreted but, using more a realistic assumption. Namely, we take the partition functions as in the case of the isotherm of d Arcy and Watt [20], but assume that the value of V for the multilayer adsorption appearing in the (5) is equal to the sum of the number of adsorbed water molecules on the Langmuir s and Henry s sites ... 

Ithough knowledge-based potentials are most popular, it is also possible to use other types potential function. Some of these are more firmly rooted in the fundamental physics of iteratomic interactions whereas others do not necessarily have any physical interpretation all but are able to discriminate the correct fold from decoy structures. These decoy ructures are generated so as to satisfy the basic principles of protein structure such as a ose-packed, hydrophobic core [Park and Levitt 1996]. The fold library is also clearly nportant in threading. For practical purposes the library should obviously not be too irge, but it should be as representative of the different protein folds as possible. To erive a fold database one would typically first use a relatively fast sequence comparison lethod in conjunction with cluster analysis to identify families of homologues, which are ssumed to have the same fold. A sequence identity threshold of about 30% is commonly... 

In this formulation, the electron density is expressed as a linear combination of basis functions similar in mathematical form to HF orbitals. A determinant is then formed from these functions, called Kohn-Sham orbitals. It is the electron density from this determinant of orbitals that is used to compute the energy. This procedure is necessary because Fermion systems can only have electron densities that arise from an antisymmetric wave function. There has been some debate over the interpretation of Kohn-Sham orbitals. It is certain that they are not mathematically equivalent to either HF orbitals or natural orbitals from correlated calculations. However, Kohn-Sham orbitals do describe the behavior of electrons in a molecule, just as the other orbitals mentioned do. DFT orbital eigenvalues do not match the energies obtained from photoelectron spectroscopy experiments as well as HF orbital energies do. The questions still being debated are how to assign similarities and how to physically interpret the differences. 

To meet the point that the amount of resonance interaction in the transition state will be dependent upon the nature of the electrophile, Yukawa and Tsuno have put forward a modified equation with three parameters. The physical interpretation of such an equation is interesting, but it is not surprising that it correlates experimental data better than does the equation with two parameters. ... 

For purposes of data correlation, model studies, and scale-up, it is useful to arrange variables into dimensionless groups. Table 6-7 lists many of the dimensionless groups commonly founa in fluid mechanics problems, along with their physical interpretations and areas of application. More extensive tabulations may oe found in Catchpole and Fulford (Ind. Eng. Chem., 58[3], 46-60 [1966]) and Fulford and Catchpole (Ind. Eng. Chem., 60[3], 71-78 [1968]). 

The calculated energy differences give a good correlation with The p parameter (p = —17) is larger than that observed experimentally for proton exchange (p — 8). A physical interpretation of this is that the theoretical results pertain to the gas phase, where... 

Several structural theories of piezoelectricity [72M01, 72M02, 72A05, 74H03] have been proposed but apparently none have been found entirely satisfactory, and nonlinear piezoelectricity is not explicitly treated. With such limited second-order theories, physical interpretations of higher-order piezoelectric constants are speculative, but such speculations may help to place some constraints on an acceptable piezoelectric theory. 

Kennedy and Benedick [67K02, 68K03] were successful in carrying out difficult Hall effect measurements in germanium samples explosively loaded at the upper end of the elastic range. Nevertheless, the measurements did not provide sufficient information to develop a physical interpretation. 

Studies of the electrical and mechanical responses of ferroelectric solids under shock compression show this technical problem to be the most complex of any investigated. The combination of rate-dependent mechanical and electrical processes along with strong electromechanical coupling has clouded physical interpretation of the numerous investigations. 

Engineering constants (sometimes known as technical constants) are generalized Young s moduli, Poisson s ratios, and shear moduli as well as some other behavioral constants that will be discussed in Section 2.6. These constants are measured in simple tests such as uniaxial tension or pure shear tests. Thus, these constants with their obvious physical interpretation have more direct meaning than the components... 

Note that the definition of R is arbitrary. However, the present choice seems simplest and has a transparent physical interpretation. The work done by the system in an infinitesimal reversible transformation at constant S, N, A, s, and ayiy is given by... 

Wavefunctions by themselves can be very beautiful objects, but they do not have any particular physical interpretation. Of more importance is the Bom interpretation of quantum mechanics, which relates the square of a wavefunction to the probability of finding a particle (in this case a particle of reduced mass /r vibrating about the centre of mass) in a certain differential region of space. This probability is given by the square of the wavefunction times dx and so we should concentrate on the square of the wavefunction rather than on the wavefunction itself. 

We generally write a for a diagonal element and p for a bonded off-diagonal element in particular uq and Pec for carbon atoms and carbon-carbon conjugated bonds. The physical interpretation is that each of the r-electrons experiences an average field due to the nuclei, the cr-electrons and the remaining TT-electrons. 

The canonical MOs are convenient for the physical interpretation of the Lagrange multipliers. Consider the energy of a system with one electron removed from orbital number Ic, and assume that the MOs are identical for the two systems (eq. (3.32)). 

If there is more than one constraint, one additional multiplier term is added for each constraint. The optimization is then performed on the Lagrange function by requiring that the gradient with respect to the x- and A-variable(s) is equal to zero. In many cases the multipliers A can be given a physical interpretation at the end. In the variational treatment of an HF wave function (Section 3.3), the MO orthogonality constraints turn out to be MO energies, and the multiplier associated with normalization of the total Cl wave function (Section 4.2) becomes the total energy. 

The physical interpretation of such a state rests in the understanding that upon a color measurement at site H and time t , there is a probability <... 

Finally, using the physical interpretation of the quantum site-state coefficients ai , we can write down an explicit form for the probability functions p . Since 0 is defined by the list n, we must simply write down the probability that a measurement of the 2r states around a given site F will yield a 2r-tuple which is an element of n. We therefore get sums of products of absolute squares, with individual list elements contributing the terms and list elements... 

Despite having a simple physical interpretation, it is clear that quantum states generally have very complex formal representations. 

In order to give the Reynolds number a physical interpretation, we can look at the typical magnitudes of individual terms of the Navier-Stokes equation. Since I V Vv V /L and vVjl , we see that... 

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