Dirac delta function approximation

A. L. Crosbie and G. W. Davidson, Dirac-Delta Function Approximations to the Scattering Phase Function, Journal of Quantitative Spectroscopy and Radiative Transfer, 33, p. 391,1985. 

The application in [24] is to celestial mechanics, in which the reduced problem for consists of the Keplerian motion of planets around the sun and in which the impulses account for interplanetary interactions. Application to MD is explored in [14]. It is not easy to find a reduced problem that can be integrated analytically however. The choice /f = 0 is always possible and this yields the simple but effective leapfrog/Stormer/Verlet method, whose use according to [22] dates back to at least 1793 [5]. This connection should allay fears concerning the quality of an approximation using Dirac delta functions. 

The partition function, Z(4>y), cannot be calculated exactly. It could be rewritten using the integral representation of the functional Dirac delta function and evaluated within the saddle place approximation. The calculations lead to the following expression [36,126,128] ... 

Here s is the Laplac variable and c. (s) the Laplace transform of the inpu. When c. (t) can be approximated by a Dirac delta function, c (s) = 1 and the right hand side of Equation 8 is the Laplace transform of the solute concentration at any z. 

Since all tracer entered the system at the same time, t = 0, the response gives the distribution or range of residence times the tracer has spent in the system. Thus, by definition, eqn. (8) is the RTD of the tracer because the tracer behaves identically to the process fluid, it is also the system RTD. This was depicted previously in Fig. 3. Furthermore, eqn. (8) is general in that it shows that the inverse of a system transfer function is equal to the RTD of that system. To create a pulse of tracer which approximates to a dirac delta function may be difficult to achieve in practice, but the simplicity of the test and ease of interpreting results is a strong incentive for using impulse response testing methods. 

Another approach to approximating the scaling functions is to construct some model spectral densities. In turn, Eqs. (14)—(19) yield the approximate Fe and Fx. The choice of the Dirac delta function for T yields trivial relations... 

Once again we have introduced the Dirac delta functions in a non-relativistic approximation. The terms in (3.157) represent a correction to the Coulomb potential. 

The Feynman diagram for the simplest annihilation event shows that annihilation is possible when the two particles are Ax h/mc 10 12 5 m apart, and that the duration of the event is At h/mc2 10-21 s. The distance is the geometric mean of nuclear and atomic dimensions, which is probably not significant. The distance is so much smaller than electronic wave functions that it may be assumed to be zero in computations of annihilation rates. The time is so short that, during it, a valence electron in a typical atom or molecule moves a distance of only ao/104, so that a spectator electron can be assumed to be stationary and the annihilating electron can be assumed to disappear in zero time. Thus the calculation of annihilation rates requires the evaluation of expectation values of the Dirac delta function, and the relaxation of the daughter system (post-annihilation remnant) can be understood with the aid of the sudden approximation [4], These are both relatively simple computations, providing an accurate wave function is available. 

Dirichlet function, which is an approximation of Delta function, S x). Various approximate representations of Dirac delta function are provided in Van der Pol Bremmer (1959) on pp 61-62. This clearly shows that we recover the applied boundary condition at y = 0. Therefore, the delta function is totally supported by the point at infinity in the wave number space (which is nothing but the circular arc of Fig. 2.20 i.e. the essential singularity of the kernel of the contour integral). 

There are M different vectors composed of N, N2, , Nm components, with Dirac delta functions centered on the univariate abscissas (in each of the M directions) corresponding to the basis function set used to approximate the functional form of the multivariate NDF. The final set of N multivariate abscissas is obtained by using the following tensor prod-... 

The Gaussian quadrature algorithm introduced in Section 3.1.1 is equivalent to approximating the univariate NDF by a sum of Dirac delta functions ... 

We make two observations regarding k 6). First, as Z increases, the region in 9 where k 9) is appreciable decreases (in fact, it behaves as l/Z). Second, the integral value of this function over the entire interval of variation of 9 is approximately 1. Thus, k 9) can be thought of as an approximation to the Dirac delta-function 5 9) (in fact, it would be more correct to refer to it as a one-sided delta-function because 9 < 0). One might expect that the reason why combustion front approximations provide results similar to those with Arrhenius kinetics is exactly that the Arrhenius exponential is an approximation to the -function. If this is the case, then a different approximation to the 5-function should yield results similar to those with Arrheirius kinetics. As a simple approximation, one can use step-functions (see Figure 3)... 

Hie functions in Rg. 7.5 approximate the Dirac delta function. Draw graphs of the corresponding functions that approximate the Heaviside step function with successively increasing accuracy. 

The Forman phase correction algorithm, presented in Chap. 2, is shown in Fig. 3.6. Initially, the raw interferogram is cropped around the zero path difference (ZPD) to get a symmetric interferogram called subset. This subset is multiplied by a triangular apodization function and Fourier transformed. With the complex phase obtained from the FFT a convolution Kernel is obtained, which is used to filter the original interferogram and correct the phase. Finally the result of the last operation is Fourier transformed to get the phase corrected spectrum. This process is repeated until the convolution Kernel approximates to a Dirac delta function. 

The Dirac delta function S (jc) can be approximated by many functions, that depend on a certain... 

Here are several functions that approximate the Dirac delta function ... 

Figure 1.1 Successive approximations to the Dirac delta function (x).

Let us see how an approximation /2 = does the job of the Dirac delta function when... 

The functions in Fig. 7.5 approximate the Dirac delta function. Draw graphs of the corresponding functions that approximate the Heaviside step function with successively increasing accuracy. Quantum mechanics postulates that the present state of an undisturbed system determines its future state. Consider the special case of a system with a time-independent Hamiltonian H. Suppose it is known that at time to the state function is fo). Derive Eq. (7.101) by substituting the expansion (7.66) with g = i/i into the time-dependent Schrodinger equation (7.97) multiply the result by i/i, integrate over all space, and solve for c . 

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