Spectral representation

Spectral Representation.—As an application of the invariance properties of quantum electrodynamics we shall now use the results obtained in the last section to deduce a representation of the vacuum expectation value of a product of two fermion operators and of two boson operators. The invariance of the theory under time inversion and more particularly the fact that... 

Similar results can be obtained for the vacuum expectation value of electromagnetic field operators. The spectral representation takes the form... 

The constant b will be determined later. A (noncovariant) spectral representation for the potentials satisfying the gauge condition... 

Consider next the current operator ju(x). The correspondence principle suggests that its form is j ( ) = — (e/2)( ( )yB, (a )]. Such a form foijn(x) does not satisfy Eq. (11-477). In fact, due to covariance, the spectral representation of the vacuum expectation value of 8u(x)Av(x), where ( ) is an arbitrary four-vector, is given by... 

Spectral representation in quantum electrodynamics, 693 Spin density... 

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

For a correlated N-electron system with a non-degenerate ground state > the one-particle Green s function has the spectral representation (20,21) ... 

Parameterization of the N-Atom Problem in Quantum Mechanics. II. Coupled-Angular-Momentum Spectral Representations for Four-Atom Systems. 

Excited triplet states n) with energy E are included in the sum for the FC and SD terms, while excited singlet states contribute to the OP term. Recalling the spectral representation of the polarization propagator for zero frequency w [60]... 

Polarization propagators or linear response functions are normally not calculated from their spectral representation but from an alternative matrix representation [25,46-48] which avoids the explicit calculation of the excited states. 

MSN. 138. T. Petrosky and I. Prigogine, Complex spectral representations and singular eigenvalue problem of Liouvillian in quantum scatering, in Symposium Quantum Physics and the Universe, Waseda University, Tokyo, Japan, 1992. 

MSN. 145. T. Petrosky and I. Prigogine, Quantum chaos, complex spectral representations and time symmetry breaking, Chaos, Solitons and Fractals 4, 311-359 (1994). 

MSN.157. 1. Antoniou, 1. Prigogine, and S. Tasaki, New spectral representations of mixing dynamical systems, in Proceedings, World Congress of Nonlinear Analysts, 1992, V. Lakshmikan-tham, ed., W. de Gruyter, Berlin, 1996. 

It turns out that not only the identity matrix I but also the matrix M itself can be expressed in terms of the eigenvalues and eigenvectors. In the so-called spectral representation of M, we have... 

Figure 4.6 Spectral representation of the bath coupling, G(w), and the modulation, F, a>). (a) General quasiperiodic modulation with peaks at (b) On-off modulation with 1/tj repetition rate for Tj Tq. (c) Impulsive PM (tr-pulses), = n. (d) Monochromatic modulation, or impulsive PM, with small phase shifts, 0

Constant strain for stress relaxation tests and constant load creep tests may be conducted in simple devices. Temperature control is critical since the results are usually applied as a spectral representation for structural analysis or research purposes. Figure 8 illustrates a multistation creep tester with automated data recorders. Strain and load endurance tests are conducted in similar devices, but the conditions existing at failure and time to failure are normally the only data required. The endurance tests are used frequently to supplement the constant displacement rate tests for routine evaluation. 

Figure 13.2. (a) Profile scan and (b) spectral representation in a typical benchtop LC-MS. 

From the completeness relations we now obtain the spectral representation... 

In complete accord with a simple numerical evaluation, / — Pmaxr/D0 and the magnitude of DQ decreases the turbulent diffusion is independent of the molecular diffusion coefficient. Let us carefully consider the structure of the quantity r = vX. It is obvious that in a turbulent flow we cannot directly determine a quantity which is linear in the fluctuation velocity. It is no accident that the square of the velocity figured in the original equation. It is precisely the mean square of the velocity and its spectral representation that may be determined in a turbulent flow. Therefore, consistently performing all the calculations, we obtain... 

Visualization of results, their statistical validity, and analysis of results are becoming increasingly important. Several techniques are available to represent data and compact representations are increasingly being made by images. The ease of visualization, however, must always be tempered with the need to present data completely and in an accurate manner. In particular, two areas are of relevant important. The first is two-dimensional spectral representation to better understand spectral structure while the second is more useful for imaging. 

Because they give access to the spectral representation of the signal, frequency-domain techniques are well suited for formant modification. The first step in frequency-domain formant modification techniques consists of obtaining a estimation of the spectral envelope. Based of the short-time representation of the signal, it. is possible to derive a spectral envelope function using a variety of different techniques. If the pitch of the signal is available, the short-time Fourier spectmm is searched for local maxima located around harmonic frequencies, then an envelope can be obtained by joining the local... 

To find the rate of the non-radiative transitions caused by the quadratic nondiagonal vibronic interaction, one needs to solve equation (27), where Di(t) in the spectral representation is given by equation (29),... 

The idea behind PCA is that a spectral dataset of mixtures of the same components can be expressed as a linear combination of a small set of spectral representations. This is most easily understood if we consider a set of mixture spectra for three components. Assuming that there is no noise and that the mixture spectra are simply the sum of absorptivity spectra for the pure components, each of the mixture spectra can... 

The Green s function of electron in a crystal has the following spectral representation [9,10] ... 

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