Negative frequencies

Secondly, y/mariefi) is analytic, that is the corresponding band pass filter has no negative frequencies. This feature is of great help because it avoids interference in the analysis between positive and negative frequencies of the signal, which alters the representation understandability. 

Negative frequencies are physically meaningless.) Does this mean that one mass oscillates at 1.00rads and the other at /3 = 1.73rads Not exactly. Behavior depends on the initial conditions. In the special case that both masses start from rest... 

Orbital-based methods can be used to compute transition structures. When a negative frequency is computed, it indicates that the geometry of the molecule corresponds to a maximum of potential energy with respect to the positions of the nuclei. The transition state of a reaction is characterized by having one negative frequency. Structures with two negative frequencies are called second-order saddle points. These structures have little relevance to chemistry since it is extremely unlikely that the molecule will be found with that structure. 

The primary means of verifying a transition structure is to compute the vibrational frequencies. A saddle point should have one negative frequency. The vibrational motion associated with this negative frequency is the motion going... 

If there are negative frequencies in an IR spectrum, it is a sign that you are not at a minimum energy structure. A valid minimum energy structure possesses only positive frequencies. 

Since E0, the vacuum energy, must for obvious physical reasons, be the lowest energy of the negaton-positon field system, En — E0 s 0, so that Eq. (10-247) indicates that for x0 > x 0, GA contains only positive frequencies. Similarly one verifies that (under the assumption of a stable vacuum state, the state of lowest energy) GA for 0 < x Q contains only negative frequencies. 

Figure 1.33 The underlying principle of the Redfield technique. Complex Fourier transformation and single-channel detection gives spectrum (a), which contains both positive and negative frequencies. These are shown separately in (b), corresponding to the positive and negative single-quantum coherences. The overlap disappears when the receiver rotates at a frequency that corresponds to half the sweep width (SW) in the rotating frame, as shown in (c). After a real Fourier transformation (involving folding about n = 0), the spectrum (d) obtained contains only the positive frequencies.

The radial frequency co of a periodic function is positive or negative, depending on the direction of the rotation of the unit vector (see Fig. 40.5). co is positive in the counter-clockwise direction and negative in the clockwise direction. From Fig. 40.5a one can see that the amplitudes (A jp) of a sine at a negative frequency, -co, with an amplitude. A, are opposite to the values of a sine function at a positive frequency, co, i.e. = Asin(-cor) = -Asin(co/) = This is a property of an antisymmetric function. A cosine function is a symmetric function because A -Acos(-co/) = Acos(cor) = A. (Fig. 40.5b). Thus, positive as well as negative... 

In Section 40.3.4 we have shown that the FT of a discrete signal consisting of 2N + 1 data points, comprises N real, N imaginary Fourier coefficients (positive frequencies) and the average value (zero frequency). We also indicated that N real and N imaginary Fourier coefficients can be defined in the negative frequency domain. In Section 40.3.1 we explained that the FT of signals, which are symmetrical about the / = 0 in the time domain contain only real Fourier coefficients. 

The asterisk designates the complex conjugate. Moreover, we note that the above Eqs. 2.46 and 2.47 imply positive as well as negative frequencies. In some physics applications, an appearance of negative frequencies may be confusing only positive frequencies may have physical meaning. In such cases one may rewrite the above inverse tranform in terms of positive frequencies, using a well-known relationship between the complex exponential function and the sine and cosine functions. 

For most treatments, the spectral density, J(a>), Eq. 2.86, also referred to as the spectral profile or line shape, is considered, since it is more directly related to physical quantities than the absorption coefficient a. The latter contains frequency-dependent factors that account for stimulated emission. For absorption, the transition frequencies ojp are positive. The spectral density may also be defined for negative frequencies which correspond to emission. 

Early attempts have modeled the positive frequency wing (v > 0) with the help of a Lorentzian, Eq. 3.15, because of a perceived similarity of the observed induced lines with the Lorentzian no theoretical justification was pretended. The negative frequency wing may then be described by exp (—hcv/kT) times that Lorentzian so that Eq. 3.18 is satisfied [215, 188, 414, 411]. Systematic deviations from this model were, however, noticed in the wings [75, 244]. Nevertheless, beautiful analyses of various rotational and rotovibrational induced bands were thus possible and significant new knowledge concerning the role of overlap and multipole induction, double transitions, etc., was obtained in this way [422],... 

In an attempt to model the spectral functions of rare gas mixtures, Fig. 3.2, it was noted that a Gaussian function with exponential tails approximates the measurements reasonably well [75], about as well as the Lorentzian core with exponential tails. Two free parameters were chosen such that at the mending point a continuous function and a continuous derivative resulted the negative frequency wing was again chosen as that same curve, multiplied by the Boltzmann factor, to satisfy Eq. 3.18. Subsequent work retained the combination of a Lorentzian with an exponential wing and made use of a desymmetrization function [320],... 

If bound state effects are suppressed, the classical profile peaks at zero frequency where it has a zero slope the classical profile is symmetric in frequency. The quantum profile, on the other hand, peaks at somewhat higher frequencies and has a logarithmic slope of h/2kT near zero frequency. At positive frequencies, the quantum profile is more intense than the classical profile, but at not too small negative frequencies the opposite is true. These facts are related to the different symmetries of these profiles, which we examine in the next subsection. We note that various procedures have been proposed to correct classical profiles somehow so that these simulate the symmetry of quantum profiles. 

At high temperatures, vibrational states must also be included in the partition sum above. The nuclear weights are gj for hydrogen we have, for example, gj = 1 for even j, and gj = 3 for odd j. However, we mention that in low-temperature laboratory measurements as well as in astrophysical applications, para-H2 and ortho-H2 abundances may actually differ from the proportions characteristic of thermal equilibrium (Eq. 6.53). In such a case, at any fixed temperature T, one may account for non-equilibrium proportions by assuming gj values so that the ratio go/gi reflects the actual para to ortho abundance ratio. Positive frequencies correspond to absorption, but the spectral function g(co T) is also defined for negative frequencies which correspond to emission. We note that the product V g a> T) actually does not depend on V because of the reciprocal F-dependence of Pt, Eq. 6.52. 

Figure 3. Field-matter interactions for a pair of electronic states. The zero and first excited vibrational levels are shown for each state (A). The fields are resonant with the electronic transitions. A horizontal bar represents an eigenstate, and a solid (dashed) vertical arrow represents a single field-matter interaction on a ket (bra) state. (See Refs. 1 and 54 for more details.) A single field-matter interaction creates an electronic superposition (coherence) state (B) that decays by electronic dephasing. Two interactions with positive and negative frequencies create electronic populations (C) or vibrational coherences either in the excited (D) or in the ground ( ) electronic states. In the latter cases (D and E) the evolution of coherence is decoupled from electronic dephasing, and the coherences decay by the vibrational dephasing process.

One often finds Sj2 = 2el in the literature. The factor 2 corresponds to the contributions of positive and negative frequencies. 

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