Fourier transform representation

The first term on the right-hand side is the time-independent expectation value, whereas the second, third, and fourth terms describe the linear, quadratic, and cubic response to the perturbation, respectively. The Fourier transformed representations are given by... 

If there is no restriction to the possible values of the phase angle and if we use the Fourier transform representation of 8i0 - 0 ), we find... 

The general task is to trace the evolution of the third order polarization of the material created by each of the above 12 Raman field operators. For brevity, we choose to select only the subset of eight that is based on two colours only—a situation that is common to almost all of the Raman spectroscopies. Three-colour Raman studies are rather rare, but are most interesting, as demonstrated at both third and fifth order by the work in Wright s laboratory [21. 22. 23 and 24]. That work anticipates variations that include infrared resonances and the birth of doubly resonant vibrational spectroscopy (DOVE) and its two-dimensional Fourier transform representations analogous to 2D NMR [25]. 

The shape of the particle can be given by the traditional sphericity factor, fractal analysis, or by Fourier transform representations. The latter is a bit involved, requiring several coefficients for complex definition. Fractal analysis is receiving more attention of late in representing the shape of the particles handled. Chapter 1, Particle Characterization and Dynamics, presents a more complete description of shape analysis. 

At this point it will be necessary to employ the Fourier transform representation of the -function, which is derived in Appendix G this representation allows us to identify the quantity in square brackets in the last expression with a -function in momentum space,... 

The potential energy part is diagonal in the coordinate representation, and we drop the hat indicating an operator henceforth. The kinetic energy part may be evaluated by transfonning to the momentum representation and carrying out a Fourier transform. The result is... 

In the further manipulations the site representation will be used for convenience. The Fourier transform with respect to time of a single-exciton retarded Green s function (t) of a system under the Hamiltonian in the site representation for exciton coordinates and Fock s representation for phonon coordinates is written as... 

Fig. 3. A block diagram schematic representation of a Fourier transform nmr spectrometer, ie, a superconducting magnetic resonance system.

Foldy, L, L., 497,498,536,539 Foldy-Wouthuysen representation, 537 "polarization operator in, 538 position operator in, 537 Ford, L. R., 259 Four-color problem, 256 Fourier transforms of Schrodinger operators, 564... 

When performing optical simulations of laser beam propagation, using either the modal representation presented before, or fast Fourier transform algorithms, the available number of modes, or complex exponentials, is not inhnite, and this imposes a frequency cutoff in the simulations. All defects with frequencies larger than this cutoff frequency are not represented in the simulations, and their effects must be represented by scalar parameters. 

For a pure state density operator, the Fourier transform of this double-time Green s function yields the spectral representation of the propagator (21)... 

Often the actions of the radial parts of the kinetic energy (see Section IIIA) on a wave packet are accomplished with fast Fourier transforms (FFTs) in the case of evenly spaced grid representations [24] or with other types of discrete variable representations (DVRs) [26, 27]. Since four-atom and larger reaction dynamics problems are computationally challenging and can sometimes benefit from implementation within parallel computing environments, it is also worthwhile to consider simpler finite difference (FD) approaches [25, 28, 29], which are more amenable to parallelization. The FD approach we describe here is a relatively simple one developed by us [25]. We were motivated by earlier work by Mazziotti [28] and we note that later work by the same author provides alternative FD methods and a different, more general perspective [29]. 

Figure 3.4 Schematic representation of the steps involved in obtaining a two-dimensional NMR spectrum. (A) Many FIDs are recorded with incremented values of the evolution time and stored. (B) Each of the FIDs is subjected to Fourier transformation to give a corresponding number of spectra. The data are transposed in such a manner that the spectra are arranged behind one another so that each peak is seen to undergo a sinusoidal modulation with A second series of Fourier transformations is carried out across these columns of peaks to produce the two-dimensional plot shown in (C).

At the end of the 2D experiment, we will have acquired a set of N FIDs composed of quadrature data points, with N /2 points from channel A and points from channel B, acquired with sequential (alternate) sampling. How the data are processed is critical for a successful outcome. The data processing involves (a) dc (direct current) correction (performed automatically by the instrument software), (b) apodization (window multiplication) of the <2 time-domain data, (c) Fourier transformation and phase correction, (d) window multiplication of the t domain data and phase correction (unless it is a magnitude or a power-mode spectrum, in which case phase correction is not required), (e) complex Fourier transformation in Fu (f) coaddition of real and imaginary data (if phase-sensitive representation is required) to give a magnitude (M) or a power-mode (P) spectrum. Additional steps may be tilting, symmetrization, and calculation of projections. A schematic representation of the steps involved is presented in Fig. 3.5. 

Figure 5.47 Two-dimensional exchange spectrum of N,Af-dimethylacetamide and its generation, (a) The first set of spectra results from the first series of Fourier transformations with respect to The modulation of signals as a function of t is observed, (b) The second set of spectra is obtained by the second series of Fourier transformations. The unmodulated signals appear on the diagonal at (v, v ), (vx, Vx), and (v, v ), whereas the modulations due to exchange show up as crosspeaks on either side of the diagonal at (vx, Vx) and (vx, Vx). (c) A contour plot representation of (b). (Reprinted from Science 232, A. Bax, et ai, 960, copyright (1986), with permission from Science-AAAS, c/o Direct Partners Int., P.O. Box 599, 1200 AN Hilversum, The Netherlands)...

The above integrals are most conveniently reduced if lrl (resp. Ir-r l )is substituted by the inverse Fourier transform of [ lrl l]7 (p) (resp. [ lr-r h ]7 (p)). The steps for the final expression of the nuclear term and the electron-electron repulsion term in p-representation are summarized helow ... 

The calculation of e in momentum space is analogous to that in position space. Starting with the r-representation, and expressing the quantity F(r)(pi(r) as the inverse Fourier transform of [F(r) (pi(r)]T(p), one easily finds that ... 

Again, this relation arises from the representation of a particle by a wave packet and is a property of Fourier transforms. 

The inverse Fourier transform then gives an integral representation of the delta function... 

An on-flow experiment is now carried out. 50 pi of a solution of the product mixture (5 mg in 5 mL solvent) are injected and the NMR proton signal accumulation started simultaneously. The time taken for the chromatogram is 17 min. During this time a total of 128 proton NMR spectra are recorded, each with eight scans, i.e. an FID is accumulated approximately every 7 sec. After the Fourier transformation we obtain a two-dimensional representation (Fig. 33) of the on-flow experiment. 

Fig. 8.32. Two-dimensional Fourier transformation applied to a rectangle function shown in original 3D representation (a) and 2D contour plot (b) and as Fourier transforms (c,d), (according to Danzer et al. [2001])...

IR dichroism has also been particularly helpful in this regard. Of predominant interest is the orientation factor S=( 1/2)(3—1) (see Chapter 8), which can be obtained experimentally from the ratio of absorbances of a chosen peak parallel and perpendicular to the direction in which an elastomer is stretched [5,249]. One representation of such results is the effect of network chain length on the reduced orientation factor [S]=S/(72—2 1), where X is the elongation. A comparison is made among typical theoretical results in which the affine model assumes the chain dimensions to change linearly with the imposed macroscopic strain, and the phantom model allows for junction fluctuations that make the relationship nonlinear. The experimental results were found to be close to the phantom relationship. Combined techniques, such as Fourier-transform infrared (FTIR) spectroscopy combined with rheometry (see Chapter 8), are also of increasing interest [250]. 

Now, we may recall the representation III of the autocorrelation function because its Fourier transform leads to the well-known Franck-Condon progression of delta Dirac peaks appearing in the pioneering work of Marechal and Witkowski [7]. In this representation III, the general autocorrelation function (2) takes the form... 

By Fourier transform of the representation III of the undamped adiabatic autocorrelation function (49), one obtains the Franck-Condon progression... 

We may recall and emphasize that the autocorrelation function obtained in the three representations I, II, and III must be equivalent, from the general properties of canonical transformation which must leave invariant the physical results. Thus, because of this equivalence, the spectral density obtained by Fourier transform of (43) and (45) will lead to the same Franck-Condon progression (51). 

Figure 3. Numerical equivalence between the three representations, I, II, and III. Within the adiabatic approximation, this figure shows the numerical equivalence between the Fourier transforms of G given by Eq. (44), Gu given by Eq. (46) and Gm given by Eq. (49).

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