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Linear superposition

Having represented the individual rich streams, we are now in a position to construct the rich composite stream. A rich composite stream represents the cumulative mass of the pollutant lost by all the rich streams. It can be readily obtained by using the diagonal nile for superposition to add up mass in the overlapped regions of streams. Hence, the rich composite stream is obtained by applying linear superposition to all the rich streams. Figure 3.4 illustrates this concept for two rich streams. [Pg.50]

Thus our arbitrary wave-front error can be described as a linear superposition of Zemike polynomials over the unit circle (although for a completely... [Pg.42]

When a system responds in proportion to the applied forces the system is said to be linear. When the response is linear, one can determine the effect of multiple forces on a system by adding the responses of individual force-response systems. This is the principle of linear superposition. [Pg.54]

Easy availability of ultrafast high intensity lasers has fuelled the dream of their use as molecular scissors to cleave selected bonds (1-3). Theoretical approaches to laser assisted control of chemical reactions have kept pace and demonstrated remarkable success (4,5) with experimental results (6-9) buttressing the theoretical claims. The different tablished theoretical approaches to control have been reviewed recently (10). While the focus of these theoretical approaches has been on field design, the photodissociation yield has also been found to be extremely sensitive to the initial vibrational state from which photolysis is induced and results for (11), HI (12,13), HCl (14) and HOD (2,3,15,16) reveal a crucial role for the initial state of the system in product selectivity and enhancement. This critical dependence on initial vibrational state indicates that a suitably optimized linear superposition of the field free vibrational states may be another route to selective control of photodissociation. [Pg.263]

Since the partial differential equation (2.6) is linear, any linear superposition of solutions is also a solution. Therefore, the most general solution of equation (2.6) for a time-independent potential energy V(x) is... [Pg.48]

Stabilization of Ru based oxides by valve metal oxides has not been studied in such detail using photoelectron spectroscopy. The most common compositions, however, with relatively high valve metal content, are not in favor of formation of a solid solution. Studies of the phase formation in Ru/Ti mixed oxides has shown [49] that homogeneous solutions are formed for compositions with Ru < 2% or Ru > 98% (see Section 3.1.1). Therefore electrodes with other compositions are better described as physical mixtures and the electrochemical behaviour is most likely that of a linear superposition of the single components. It has to be considered, however, that the investigations performed by Triggs [49] concern thermodynamic equilibrium conditions. If, by means of the preparation procedure, thermodynamic equilibrium is... [Pg.108]

Since the one-dimensional wave equation is linear, the general solution periodic in x with period 2n is the linear superposition... [Pg.48]

Since H is Hermitian, the eigenvectors Vj of H form a complete orthonormal set and the vector representing a general state at t = 0 may be expressed as a linear superposition of these eigenvectors, (0) = CjVj, ... [Pg.196]

As shown in Fig. 5.4, the flow domain can be denoted by 2 with inlet streams at Ain boundaries denoted by 3 2, (/el,..., Ain). In many scalar mixing problems, the initial conditions in the flow domain are uniform, i.e., cc(x, 0) = 40). Likewise, the scalar values at the inlet streams are often constant so that cc(x e 3 2, t) = c(f for all / e 1,..., Nm. Under these assumptions,38 the principle of linear superposition leads to the following relationship ... [Pg.176]

Applying the principle of linear superposition, ipc can be written as42... [Pg.177]

It is instructive to start with the excitation spectrum in the case of the ordinary 2SC phase when dfi = 0. With the conventional choice of the gap pointing in the anti-blue direction in color space, the blue quarks are not affected by the pairing dynamics, and the other four quasi-particle excitations are linear superpositions of ur>g and dr(J quarks and holes. The quasi-particle is nearly identical with a quark at large momenta and with a hole at small momenta. We represent the quasi-particle in the form of Q(quark, hole), then the four quasiparticles can be represented explicitly as Q(ur,dg), Q(ug, dr), Q(dr,ug) and Q(dg,ur). When S/i = 0, the four quasi-particles are degenerate, and have a common gap A. [Pg.230]

The pair potential of colloidal particles, i.e. the potential energy of interaction between a pair of colloidal particles as a function of separation distance, is calculated from the linear superposition of the individual energy curves. When this was done using the attractive potential calculated from London dispersion forces, Fa, and electrostatic repulsion, Ve, the theory was called the DLVO Theory (from Derjaguin, Landau, Verwey and Overbeek). Here we will use the term to include other potentials, such as those arising from depletion interactions, Kd, and steric repulsion, Vs, and so we may write the total potential energy of interaction as... [Pg.49]

It should be remembered that ox = 0 while t < t. This linear superposition of stresses can be generalised to any number of applied stresses ... [Pg.120]

THE BERENS-HQPFENBERG MODEL. The Berens and Hopfenberg model considers the sorption process in glassy polymers as a linear superposition of independent contributions of a rapid Fickian diffusion into pre-existing holes or vacancies (adsorption) and a slower relaxation of the polymeric network (swelling).(lS) The total amount of sorption per unit weight of polymer may be expressed as... [Pg.152]

In principle, the time evolution of a particular linear superposition on the molecular base states will reflect a chemical process via the changes shown by the amplitudes. This represents a complete quantum mechanical representation of the chemical processes in Hilbert space. The problem is that the separability cannot be achieved in a complete and exact manner. One way to introduce a model that is able to keep as much as possible of the linear superposition principle is to use generalized electronic diabatic base functions. [Pg.182]

Separability between electronic and nuclear states is fundamental to get a description in terms of a hierarchy of electronic and subsidiary nuclear quantum numbers. Physical quantum states, i.e. wavefiinctions 0(q,Q), are non-separable. On the contrary, there is a special base set of functions Pjt(q,Q) that can be separable in a well defined mode, and used to represent quantum states as linear superpositions over the base of separable molecular states. For the electronic part, the symmetric group offers a way to assign quantum numbers in terms of irreducible representations [17]. Space base functions can hence be either symmetric or anti-symmetric to odd label permutations. The spin part can be treated in a similar fashion [17]. The concept of molecular species can be introduced this is done at a later stage [10]. Molecular states and molecular species are not the same things. The latter belong to classical chemistry, the former are base function in molecular Hilbert space. [Pg.182]

The ( )j(q)Qm (Q) is a set of linearly independent functions the Qm(Q) functions are not orthogonal in Q-space for arbitrary electronic states the overlap integrals Jd Q Qm(Q) Q m (Q) are the well known Franck-Condon factors. The hypothesis is that an arbitrary quantum molecular state is given by the linear superposition jus as in the general case ... [Pg.184]

Once the electronic diabatic base set is obtained, the quantum state is a linear superposition where the PCB configuration enters parametrically as Ck amplitudes ... [Pg.187]

The electron-phonon operator is a tensor product between the electronic dipole and the nuclear dipole operators. A mixing between the AA and BB via the singlet-spin diradical AB state is possible now. A linear superposition of identical vibration states in AA and BB is performed by the excited state diradical AB. If the system started at cis state, after coupling may decohere by emission of a vibration photon in the trans state furthermore, relaxation to the trans... [Pg.190]

No one of the equations introduced here are defined as in the standard Bom-Oppenheimer approach. The reason is that electronic base functions that depend parametrically on the geometry of the sources of external potential are not used. The concept of a quantum state with parametric dependence is different. This latter is a linear superposition the other are objects gathered in column vectors. [Pg.193]

Finally, the rules of angular momentum construction can be made as if the system had spherical symmetry. The reason is that the invariance to rotation of the I-frame leads to angular momentum conservation. Once all base states have been constructed, the dynamics is reflected on the quantum state that is a linear superposition on that base. As the amplitudes change in time, motion of different kinds result. [Pg.193]

In linear superposition, the method is literally that of adding components. When treating the optics of coherent light, for example, the instantaneous values of the field vectors are superimposed. Incoherent light, on the other hand, requires us to deal with the time-averaged square of the field. In nonlinear optics, superposition breaks down as it does in other nonlinear systems. Even when it does not hold exactly, however, superposition is often useful as a first-order approximation. [Pg.18]

Knowing that the better nonlinear constrained methods are now available, why have researchers generally been reluctant to accept them Perhaps the linear approach has an attraction that is not related to performance. Early in a technical career the scientist-engineer is indoctrinated with the principles of linear superposition and analysis. Indeed, a rather large body of knowledge is based on linear methods. The trap that the linear methods lay for us is the existence of a beautiful and complete formalism developed over the years. Why complicate it by requiring the solution to be physically possible ... [Pg.96]

We see from Fig. 2.5 that the Gaussian wave packet has its intensity, F 2, centred on x0 with a half width, W, whereas (k) 2 is centred on k0 with a half width, 1/W. Thus the wave packet, which is centred on x0 with a spread Ax — W, is a linear superposition of plane waves whose wave vectors are centred on k0 with a spread, A = jW. But from eqn (2.8), p = Hk. Therefore, this wave packet can be thought of as representing a particle that is located approximately within Ax = W of x0 with a momentum within Ap = h/W of po = hk0. If we try to localize the wave packet by decreasing W, we increase the spread in momentum about p0. Similarly, if we try to characterize the particle with a definite momentum by decreasing 1/W, we increase the uncertainty in position. [Pg.26]

We note the very important property that this equation is linear so that if % and Pa are solutions, so also is + A2x 2t generalizing the linear superposition of plane waves we saw earlier. [Pg.30]

The case of a decomposable matrix (2.6) merely means that one has two non-interacting systems, governed by two M-equations with matrices A and B, respectively. A non-trivial example is a system in which all transitions conserve energy each energy shell E has its own M-equation and its own stationary distribution . The stationary solutions of the total M-equation are linear superpositions of them with arbitrary coefficients nEi... [Pg.102]

The essential principle of coherent control in the continuum is to create a linear superposition of degenerate continuum eigenstates out of which the desired process (e.g., dissociation) occurs. If one can alter the coefficients a of the superposition at will, then the probabilities of processes, which derive from squares of amplitudes, will display an interference term whose magnitude depends upon the a,. Thus, varying the coefficients a, allows control over the product properties via quantum interference. This strategy forms the basis for coherent control scenarios in which multiple optical excitation routes are used to dissociate a molecule. It is important to emphasize that interference effects relevant for control over product distributions arise only from energetically degenerate states [7], a feature that is central to the discussion below. [Pg.296]


See other pages where Linear superposition is mentioned: [Pg.225]    [Pg.411]    [Pg.203]    [Pg.139]    [Pg.106]    [Pg.87]    [Pg.455]    [Pg.2]    [Pg.177]    [Pg.178]    [Pg.185]    [Pg.190]    [Pg.190]    [Pg.193]    [Pg.48]    [Pg.50]    [Pg.50]    [Pg.148]    [Pg.112]    [Pg.779]    [Pg.271]   
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See also in sourсe #XX -- [ Pg.30 ]

See also in sourсe #XX -- [ Pg.61 ]




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