Complex quantities

In addition, it can occasionally be useful to regard some physical parameter appearing in the theoi y as a complex quantity and the wave function to possess analytic properties with regard to them. This formal procedure might even include fundamental constants like e, h, and so on. 

Eddy current and hysteresis losses are complex quantities and can be estimated in a laboratory, on no load, in the form of power input, le.ss and friction and windage losses etc. as shown in Section 11.5. Based... 

The transfer coefficient yand the characteristic impedance Z are important parameters of the pipeline and consist of complex quantities, in contrast to the analogous data in Section 24.4. Substituting Z and Y from Eqs. (23-1) to (23-4) leads to ... 

If the rate of sweep through the resonance frequeney is small (so-called slow passage), a steady-state solution, in which the derivatives are set to zero, is ob-tained. The result expresses M,., and as funetions of cu. These magnetization components are not actually observed, however, and it is more useful to express the solutions in terms of the susceptibility, a complex quantity related to the magnetization. The solutions for the real (x ) and imaginary (x") components then are... 

Wavefunctions are often complex quantities, and we have to be careful to distinguish a wavefunction vp from its complex conjugate P. For most of this text, wavefunctions will be real quantities and so we can drop the complex conjugate sign without lack of mathematical rigour. 

I have included the modulus bars in IV (r)p because wavefunctions can be complex quantities. For most of this and subsequent chapters, I will assume that we are dealing with real wavefunctions. 

Note that the term y in Eqs. 2-15 and 2-16 has a different significance than that in Eq. 2-14. In the first equation it is based on a concept of relaxation and in the others on the basis of creep. In the literature, these terms are respectively referred to as a relaxation time and a retardation time, leading for infinite elements in the deformation models to complex quantities known as relaxation and retardation functions. One of the principal accomplishments of viscoelastic theory is the correlation of these quantities analytically so that creep deformation can be predicted from relaxation data and relaxation data from creep deformation data. 

Visco-elastic substances can be described with the spring/shock-absorber model of Kevin and Voigt, and have phase displacements of 0° to 90°. In analogy to other time dependent processes in physics, the oscillation tests are evaluated with complex arithmetics. Obtained are the complex quantities ... 

Before closing this section, it is worth mentioning that the hyperpolarizability tensors are complex quantities usually given in the old cgs system of units of esu (electrostatic units). The transformation into the International System is readily obtained with the relationship ... 

For low enough symmetries, both Bk and B q coefficients will be present in Equation 1.15, so that Equation 1.17 will be, in those cases, complex quantities. We finally note that the coefficients are transformed into CF parameters by multiplying them by the radial parts of the wave functions, represented by Rn/(r), on which the tensor operators do not act. 

The most important and fundamental difference between quantum mechanics and classical mechanics is the appearance of complex quantities as an essential ingredient of the former. 

The magnitude of a complex quantity can be obtained from the product... 

Then the thermal propagator is represented by a sum over free propagators each one displaced in time by the pure complex quantity i(3l. Actually this result can be seen as describing a scalar field constrained in the time axes, now considered as a sum of complex quantities. Each term G0(x — x — ifiln) in G(x — xr-,/3)n is but the contribution of images reflected on the walls placed at Im(t) = 0 and Im(t) = f3. This interpretation is reinforced by a similar expression that can be derived if we consider the field within a slab of two parallel planes, such that for instance one at z = 0 and the other at z = L (J.C. da Silva et.al., 2002). 

The conductivity of an ionic conductor can be assessed by direct current (dc) or alternating current (ac) methods. Direct current methods give the resistance R and the capacitance C. The corresponding physical quantity when ac is applied is the impedance, Z, which is the total opposition to the flow of the current. The unit of impedance is the ohm (fl). The impedance is a function of the frequency of the applied current and is sometimes written Z(to) to emphasize this point. Impedance is expressed as a complex quantity ... 

A delay occurs between attainment of supersaturation and detection of the first newly created crystals in a solution, and this so-called induction period, t, is a complex quantity that involves both nucleation and growth components. If it is assumed that /, is essentially concerned with nucleation, that is t, ot 1 /J, then Mullin 3 has shown, from equation 15.9, that ... 

Unless specifically noted, the equations below are assumed to apply to transition state quantities ( ) as well as to equilibrium quantities. All A quantities are then understood either as differences between activated complex quantities and reactant quantities, or between product and reactant quantities. 

In the high temperature limit where all the nuclear motions coupled to the process can be described classically, the nuclear factor is expressed in terms of only two parameters the driving force of the reaction AG°, and the whole reorganization energy X (expressions (13) and (14)). Detailed calculations carried out in the case of cytochrome c have demonstrated that AG° is a complex quantity, which depends not only on the electronic properties of the redox centers but also on those of the protein and of the surrounding solvent [100]. Usually, AG can be evaluated from measurements of redox potentials and of eventual interaction energies between the different parts of the systems (Appendix). 

Note that we need F, a complex quantity, to find the electron density. However, from the intensity we can only derive the amplitude of F. The lack of phase information requires specific methods for solution, beyond the scope of this book. 

It is a complex quantity. The phase lag ( > between the stress and the strain determines the real part G (the elastic component) and the imaginary part G" (the loss component) ... 

We have introduced the boldface notation to underline that AF is a vector in the complex plane (see Fig. 5.9), because both Fobs and Fcalc are, in general, complex quantities, as is evident from Eq. (5.6). The phase angles

further discussed in section 5.2.5. In a different notation we may write, like Eq. (5.5),... 

The fits given above are not meant to replace more accurate tabulations. Instead, they serve to bring out the essential dependence of these complex quantities on the atomic number. 

Most polymers are applied either as elastomers or as solids. Here, their mechanical properties are the predominant characteristics quantities like the elasticity modulus (Young modulus) E, the shear modulus G, and the temperature-and frequency dependences thereof are of special interest when a material is selected for an application. The mechanical properties of polymers sometimes follow rules which are quite different from those of non-polymeric materials. For example, most polymers do not follow a sudden mechanical load immediately but rather yield slowly, i.e., the deformation increases with time ( retardation ). If the shape of a polymeric item is changed suddenly, the initially high internal stress decreases slowly ( relaxation ). Finally, when an external force (an enforced deformation) is applied to a polymeric material which changes over time with constant (sinus-like) frequency, a phase shift is observed between the force (deformation) and the deformation (internal stress). Therefore, mechanic modules of polymers have to be expressed as complex quantities (see Sect. 2.3.5). 

Reorientation of dipoles in the direction of the field is characterized by a relaxation time, T. When t is small compared to the frequency of the applied voltage, instantaneous polarization occurs when t is large, the resultant polarization will be free from orientational contribution. A dispersion of polarization and hence of e, occurs when the frequency and t are close to each other and e, in this region is a complex quantity, — i . In the region of dispersion there is a phase lag between the applied field and the instantaneous polarization when the phase angle 5 is small, tan 5 = eJe,. Furthermore,... 

The attenuation may be expressed by making the wavenumber complex (this would be k — ia in eqn (6.12)), and the velocity (= w/k) may also be written as a complex quantity. This in turn corresponds to a complex modulus, so that the relationship v - /(B/p) is preserved indeed the acoustic wave equation may be written as a complex-valued equation, without the need for the extra term in (6.11). Complex-valued elastic moduli are frequency-dependent, and the frequency-dependent attenuation and the velocity dispersion are linked by a causal Kramers-Kronig relationship (Lee et al. 1990). 

Throughout this section all V are functions of z they have been printed bold in (8.20) to emphasize that they contain phase information. They are summed as complex quantities, i.e. with respect to both their amplitude and phase. But in the usual experimental implementation, where V is measured through a diode detector, the phase information of V is not available. Therefore, if the system has been calibrated to give square law detection, the measured signal may be represented as... 

In isotropic materials, and along symmetry directions in anisotropic materials, all the traction components in the third equation vanish, as does the component of A associated with the SH mode. There are then three equations for three unknowns. For the general anisotropic case the four equations can be solved to give the amplitude A4 of the reflected wave. The amplitude is a complex quantity, because there may be a change in phase upon reflection. For an incident wave at an angle d to the normal and 0 to some direction lying in the surface (usually the lowest index direction available) the reflectance function may be written... 

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