Complex representation

Furthennore, the non-oscillating component of the integrand can best be sorted out by going to the complex representation of the total field, the polarization, and the susceptibility. The mathematically pure real quantities in equation (Bl.3.2) can be written in their complex representation as follows ... 

The second way to achieve quadrahire is to introduce another field, E, (called a local oscillator) designed in frequency and wavevector to conjugate (go into quadrahire) in its complex representation with the new field of interest. Thus in the heterodyne case, the signal photons are derived fromcr. jy i. or Sj (lieterodyne) x x X... 

In using the complex representation (equation (Bl.3.4)), theyth electric field is given as... 

B1.3.2.2 THE GENERATORS FOR ALL THIRD ORDER SPECTROSCOPIES FROM THE COMPLEX REPRESENTATION OF THE FIELD... 

It is convenient to discuss the linear Jahn-Teller model in the scaled complex representation... 

In this section we represent wave packets as vectors v and refer to the individual components of v in the computer-friendly form of multidimensional arrays with indices, for example, v i,j,k), where the indices i,j, and k refer to the various degrees of freedom. These vectors may be real, such as the real vector V of the previous sections, or they may be complex representations of the... 

A complex representation of IMPS data obtained for the heterogeneous quenching of ZnTPPC -diferrocenylethane is displayed in Fig. 21(a). The semicircular response in the first quadrant corresponds to the competition between product separation and back electron transfer, while the lower quadrant is determined by the i uQi constant. The i uQi attenuation limited the frequency range to less than 1 kHz. Equation (45) describes the experimental spectra at various Galvani potential differences [solid lines in Fig. 21(a)],... 

Complex representations, multidegenerate nonlinear coupling, higher order coupling, 243 -244... 

Nonlinear coupling, multidegenerate conditions higher order coupling, complex representations, 243-244 molecular systems, 233-249 adiabatic-to-diabatic transformation, 241— 242... 

This version uses a more complex representation, not detailed here, to maintain the events so that they are indexed directly by their date ranges. This data structure is encapsulated behind an interface called EventContainer that does all the real work. 

The vector Fc is a complex representation of the real field F. If all our operations on time-harmonic fields are linear (e.g., addition, differentiation, integration), it is more convenient to work with the complex representation. The reason this may be done is as follows. Let be any linear operator we can operate on the field (2.10) by operating on the complex representation (2.11) and then take the real part of the result ... 

Note that there is a degree of arbitrariness associated with the complex representation of a real field F could just as easily have been written F - Re(F, where F = C exp(/to/) and the asterisk denotes the complex conjugate. Thus, there are two possible choices for the time-dependent factor in u complex representation of a time-harmonic field exp(/co/) and exp( — iat). It mukcs no difference which choice is made the quantities of physical interest arc ttlwuys real. But once a sign convention has been chosen it must be used conniMcntly in all analysis. We shall take the time-dependent factor to be exp(-/u>/) this is the convention found in standard books on optics (Born Mini Wolf, 1965) and electromagnetic theory (Stratton, 1941 Jackson, 1975) as... 

Equations (2.12)—(2.16) will usually be our point of departure in scattering problems. However, to avoid a cluttered notation, we shall often omit the subscript c from the complex fields. In those instances where confusion might result, the subscript will be retained, although it should usually be clear from the context if we are dealing with real fields or their complex representations. 

The electric field is taken to be time harmonic with frequency co. As in previous chapters, we shall deal with the complex representations of the real... 

Here the components of excited state J are expressed in a representation that diagonalizes the spin-orbit operator. In general, this will be a complex representation. The principle of spectroscopic stability can again be used to express the components of Jin a representation that we denote jM. This representation is made up of space and spin parts where the spin part diagonalizes the spin operator. 

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