Square modulus

The computed CWT leads to complex coefficients. Therefore total information provided by the transform needs a double representation (modulus and phase). However, as the representation in the time-frequency plane of the phase of the CWT is generally quite difficult to interpret, we shall focus on the modulus of the CWT. Furthermore, it is known that the square modulus of the transform, CWT(s(t)) I corresponds to a distribution of the energy of s(t) in the time frequency plane [4], This property enhances the interpretability of the analysis. Indeed, each pattern formed in the representation can be understood as a part of the signal s total energy. This representation is called "scalogram". 

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

Flence, all probabilities given by the squared modulus of the wavefiinction decrease as exp[-(F yD)t] with time, that is... 

In this expression, factors that describe the incident and scattered projectile are separated from the square modulus of an integral that describes the role of the target in detemiining the differential cross section. The temi preceding the... 

Note [240] that the phase in Eq. (13) is gauge independent. Based on the above mentioned heuristic conjecture (but fully justified, to our mind, in the light of our rigorous results), Resta noted that Within a finite system two alternative descriptions [in temis of the squared modulus of the wave function, or in temis of its phase] are equivalent [247]. 

In other words, the output power is the area under the product of the Fourier transform of Rx(r) and the squared modulus of the system function H(f). 

We shall in this chapter be most concerned with the following example of Hilbert space. Each element /> is a complex-valued numerical function f(x) of one or more continuous variables represented collectively by the symbol x, such that the integral of its square modulus exists ... 

Density functions can be obtained up to any order from the manipulation of the Slater determinant functions alone as defined in section 5.1 or from any of the linear combinations defined in section 5.2. Density functions of any order can be constructed by means of Lowdin or McWeeny descriptions [17], being the diagonal elements of the so called m-th order density matrix, as was named by Lowdin the whole set of possible density functions. For a system of n electrons the n-th order density function is constructed from the square modulus of any n-electron wavefunction attached to the n-electron system somehow. 

Further degradation of the information encoded in the electron beam takes place in the recording step since the signal is proportional to the square modulus of the image wave-function, i.e. neglecting small second order terms ... 

TRXRD detects the propagation of coherent acoustic phonons as a transient change in the diffraction angles. In contrast, the atomic motions associated with coherent optical phonons modify only the Bragg peak intensity, because they do not change the barycentric positions of the crystal lattice. The Bragg peak intensity is proportional to the squared modulus of the structure factor [1,3,4] ... 

After some algebra, the probability of jumping from one state to the other becomes proportional to the square modulus of the matrix element of the dipole moment between the bare quantum states k and j (if different from zero, or to any multipole moment having non-zero matrix elements between these two states). 

In general, the squared-modulus of the vector function f(x) over the domain Sd is... 

Let us choose the initial guess for x<0)=(0.6,0.3). Successive steps produce the results shown in Table 3.11. Figure 3.10 shows the Gibbs free enthalpy of the liquid and solid mixtures together with the final result x<6)=(0.179 62,0.63106). The last column in Table 3.11 lists the squared-modulus s=fTf of the vector/as a convenient measure of convergence, o... 

The noise component of the reduced vector is the complement of to f. The squared modulus of this noise, which is the reduced squared distance (df)2 of the ith actual measurement to the point represented by its significant principal components, is... 

The excitation spectrum proves even more useful near the surface. Since anisotropic molecules at the surface of a liquid tend to orient relative to the surface tangent, one might expect the excitation spectrum to be sensitive to such orientation. For example, suppose we take the extreme case in which molecules at the surface are oriented with their transition moments perpendicular to the surface tangent. Then the only field component which can excite these molecules is the radial field at the surface. When one recalls that only the N type vector field has radial components, one expects that a calculation of the excitation spectrum of such a molecular layer will yield half as many resonant features as shown in Figure 8.4. Indeed this is the case. Figure 8.7 shows the calculated surface average of the square modulus of the radial component of the local electric field, < E er 2>J, where sr is the radial unit vector. 

In a mixed quantum-classical simulation such as a mean-field-trajectory or a surface-hopping calculation, the population probability of the diabatic state v[/ t) is given as the quasiclassical average over the squared modulus of the diabatic electronic coefficients dk t) defined in Eq. (27). This yields... 

As a consistency test of the stochastic model, one can check whether the percentage Nk t) of trajectories propagating on the adiabatic PES Wk is equal to the corresponding adiabatic population probability Pf t). In a SH calculation, the latter quantity may be evaluated by an ensemble average over the squared modulus of the adiabatic electronic coefficients [cf. Eq. (22)], that is. 

The previous section dealt with a horizontal scale change in one domain resulting in both horizontal and vertical changes in the other domain. Rayleigh s theorem, on the other hand, makes a statement about the area under the squared modulus. This integral, in fact, has the same value in both domains ... 

The two waves—one , from the common source S the other, 0 from the generator of theta waves S —overlap at the detection region. The expected intensity at the array of detectors DR, after n arrivals of particles, is given by the squared modulus of the superposition of two waves at each instant of time, summed for all n arrivals ... 

This equation allows for the fact that the density of light flux is equal to the squared modulus of the wave electric field in a semiconductor < (2) is found by solving Eq. (51). 

Fig. 3. Spin-coupled orbital (fq in benzene — throughout this review we present contour plots of the square modulus of the orbitals,

Figure 4 Isosurfaces of the square modulus of the HOMO and LUMO for Si Etjo, Sii4Hi6=C>2, Si 41 [20 f >2, Sii2Hi6>02 at the optimized geometry. The isosurfaces are plotted at 50% of their maximum amplitude.

Figure 12 Isosurfaces at fixed value (10% of max. amplitude) of the square modulus of the highest occupied (HOMO) and lowest unoccupied (LUMO) Kohn-Sham orbitals for the Si m cluster in the Si02 matrix.

R — oo. As we will show below, the partial cross sections for absorbing the photon and producing the diatomic fragment in vibrational channel n are proportional to the square modulus of the overlap of these continuum wavefunctions with the nuclear wavefunction in the electronic ground state (indicated by the shaded areas). Since the bound wavefunction of the parent molecule is rather confined, only a very small portion of the continuum wavefunctions is sampled in the overlap integral. 

The sum of the waves that result in these two experiments must equal the original plane wave incident on the particle, and the two disturbance electric fields due to diffraction are of equal magnitude, but are of opposite sign. Since the intensity is the square modulus of the electric fields, either case will produce the same answer since the sign will not affect the result. It will turn out to be simpler to calculate the scatter light intensity using the second model presented above. 

By contrast, the square modulus of the orbital angular momentum (see Eq. [22]) commutes with all three components of S , that is,... 

Phase conventions have been chosen to be consistent with those of Condon and Shortley.13 In terms of tensor operators, the square modulus of f becomes... 

As for the orbital angular momentum, the commutation relations between the Cartesian components of a general angular momentum / and its square modulus A read... 

Further, it is easily seen that the square modulus of d equals the unit matrix in two dimensions I2... 

In the early view, there are correspondence rules relating the primitive concept of state and observable to empirical reality. Observables are mapped on to the set of eigenvalues of a particular class of self-adjoint operators (e.g., Hamiltonians). The individual systems would occupy only one base state the amplitude appearing in the linear superposition in square modulus represents the probability to find one system occupying a base state when scanning the ensemble. 

The just exposed correlations then do not originate between a measuring apparatus and system being observed as a matter of fact, there is no need for an observer. The quantum state at the recording screen is totally independent from the presence of the sensitized screen element. Bearing in mind that a quantum state does not represent a particle in laboratory space and the idea that its square modulus stands for the probability of finding the material system (particle) in a volume element is foreign to the present view, the underlined result obtained from the present theoretical approach is reasonable from a physical point of view It is measurement of a wavefunction that is a basic element of the theory. 

If one is interested only in the molecular electron excitations induced by p decay, and sums up the squared modulus of the molecular matrix ele-... 

In order to find the relative probability of the molecule being excited into the nth electronic state it is sufficient to consider only the molecular part of the total matrix element [Eq. (12)], to sum the squared modulus of the molecular matrix element [Eq. (13)] over the quantum numbers of nuclear motion of the final state, and the average it over the quantum numbers of the initial state... 

As was shown in Section II, the squared modulus of the overlap integral [Eq. (A.6)] for the parent and the daughter molecules gives the probability of molecular excitations induced by / decay. 

The reflectivity and transmissivity of the resonator may be found by calculating the squared modulus of the amplitude of reflection and transmission. We find... 

There are several ways to define ionic charges. When the eigen-states of the effective one-electron Hamiltonian are expanded on a localised basis set, the probability of presence of the electrons contains a site contribution (square modulus of the projection of the wave function on the sites) and a bond term, related to the overlap of the basis functions. In standard Mulliken population analysis [16], each bond contribution is equally shared between the atom pair. The charges then depend sensitively upon the choice of the basis set and it is meaningless to compare absolute values obtained with different methods. Only charge variations within a given method bear significance. 

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