Spectral density operator

As pointed out in the previous section, the Chebyshev operator can be viewed as a cosine propagator. By analogy, both the energy wave function and the spectrum can also be obtained using a spectral method. More specifically, the spectral density operator can be defined in terms of the conjugate Chebyshev order (k) and Chebyshev angle (0) 128 132... 

Orthogonal Polynomial Expansion of the Spectral Density Operator and the Calculation of Bound State Energies and Eigenfunctions. 

T.l.B Basic Physical Observables and Corresponding Scalars T. 1. C Physical Parameters Dealing With Spectroscopy T. 1. D Scalar Functions Dealing With the Spectral Density Operators, Kets, and Density Operators T.2.A Operators that Are Not Observable T.2.B Hermitean Hamiltonians... 

For calculating bound-state energies and wave functions, A 0 and the filter becomes equivalent to the spectral density operator S Ei — H) [208]. In the case of resonances, the Green s function in Eq. (33) selects the contributions from those complex poles whose real parts lie near Ei. In what follows we consider only the filter defined in Eq. (33). 

Equation (17) is based on the fact that the spectral density operator h E - ft) projects any 2 function (or wave packet) into the space of solutions of the Schrodinger equation at the energy E. However, since the functions are needed only inside interaction region and do not have to be properly normalized, the 8( - ft) in Eqs. (17) and, consequently, (18) can be replaced by any other projector onto the solution space inside the interaction region. A convenient form for such a spectral projector related to Eq. (9) is... 

In Sec. III.A the generic stabilization method is described, and it is shown that the spectral density of the system can be computed from the eigenvalues used to obtain the stabilization diagram and their corresponding eigenfunctions. Section III.B shows how to use the spectral density operator to trivially construct N() independent solutions of the Schrodinger equation and then how to use these solutions to calculate the 5-matrix and other observables. Section III.C presents illustrative results on calculation of resonances for the HCO formyl radical. 

A. Bound State Calculations Using the Spectral Density Operator... 

Two most appealing features of this model draw so much attention to it. First, although microscopically one has very little information about the parameters entering into (5.24), it is known [Caldeira and Leggett 1983] that when the bath responds linearly to the particle motion, the operators q and p satisfying (5.24) can always be constructed, and the only quantity entering into the various observables obtained from the model (5.24) is the spectral 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)... 

The linear response theory [50,51] provides us with an adequate framework in order to study the dynamics of the hydrogen bond because it allows us to account for relaxational mechanisms. If one assumes that the time-dependent electrical field is weak, such that its interaction with the stretching vibration X-H Y may be treated perturbatively to first order, linearly with respect to the electrical field, then the IR spectral density may be obtained by the Fourier transform of the autocorrelation function G(t) of the dipole moment operator of the X-H bond ... 

Figure 5. Illustration of the equivalence between the spectral densities obtained within the adiabatic approximation and those resulting from the effective Hamiltonian procedure, using the wave operator. Common parameters a0 = 0.4, co0 = 3000cm-1, C0Oo = 150cm-1, y = 30cm-1, and T = 300 K.

Resolution can only be determined after noise evaluation of the sensor, keeping in mind that noise is related to the operating point. In practical situations different kinds of noises can be encountered Thermal, Flicker, Generation-Recombination, Shot, and others that are seen in special cases but are not so frequent. The most important parameter used for the characterization of noise devices is the Noise Spectral Density by which, through integration, it is possible to estimate the mean square value of the output voltage ... 

How does one extract eigenpairs from Chebyshev vectors One possibility is to use the spectral method. The commonly used version of the spectral method is based on the time-energy conjugacy and extracts energy domain properties from those in the time domain.145,146 In particular, the energy wave function, obtained by applying the spectral density, or Dirac delta filter operator (8(E — H)), onto an arbitrary initial wave function ( (f)(0)))1 ... 

In order to evaluate the spectral density of Eq. (35) or (38), one needs a complete basis set spanning the lattice operator space. This basis set can be obtained by taking direct products of Wigner rotation matrices,... 

As in Eq. (64), the electron spin spectral densities could be evaluated by expanding the electron spin tensor operators in a Liouville space basis set of the static Hamiltonian. The outer-sphere electron spin spectral densities are more complicated to evaluate than their inner-sphere counterparts, since they involve integration over the variable u, in analogy with Eqs. (68) and (69). The main simplifying assumption employed for the electron spin system is that the electron spin relaxation processes can be described by the Redfield theory in the same manner as for the inner-sphere counterpart (95). A comparison between the predictions of the analytical approach presented above, and other models of the outer-sphere relaxation, the Hwang and Freed model (HF) (138), its modification including electron spin... 

R is called the relaxation superoperator. Expanding the density operator in a suitable basis (e.g., product operators [7]), the a above acquires the meaning of a vector in a multidimensional space, and eq. (2.1) is thereby converted into a system of linear differential equations. R in this formulation is a matrix, sometimes called the relaxation supermatrix. The elements of R are given as linear combinations of the spectral density functions (a ), taken at frequencies corresponding to the energy level differences in the spin system. 

The indices k in the Ihs above denote a pair of basis operators, coupled by the element Rk. - The indices n and /i denote individual interactions (dipole-dipole, anisotropic shielding etc) the double sum over /x and /x indicates the possible occurrence of interference terms between different interactions [9]. The spectral density functions are in turn related to the time-correlation functions (TCFs), the fundamental quantities in non-equilibrium statistical mechanics. The time-correlation functions depend on the strength of the interactions involved and on their modulation by stochastic processes. The TCFs provide the fundamental link between the spin relaxation and molecular dynamics in condensed matter. In many common cases, the TCFs and the spectral density functions can, to a good approximation, be... 

All that remains to be done for determining the fluctuation spectrum is to compute the conditional average, Eq. (31). However, this involves the full equations of motion of the many-body system and one can at best hope for a suitable approximate method. There are two such methods available. The first method is the Master Equation approach described above. Relying on the fact that the operator Q represents a macroscopic observable quantity, one assumes that on a coarse-grained level it constitutes a Markov process. The microscopic equations are then only required for computing the transition probabilities per unit time, W(q q ), for example by means of Dirac s time-dependent perturbation theory. Subsequently, one has to solve the Master Equation, as described in Section TV, to find both the spectral density of equilibrium fluctuations and the macroscopic phenomenological equation. 

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