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Dirac function equality

Note that here bracket does not mean just any round, square, or curly bracket but specifically the symbols and > known as the angle brackets or chevrons. Then ( /l is called a bra and Ivp) is a ket, which is much more than a word play because a bra wavefunction is the complex conjugate of the ket wavefunction (i.e., obtained from the ket by replacing all f s by -i s), and Equation 7.6 implies that in order to obtain the energies of a static molecule we must first let the Hamiltonian work to the right on its ket wavefunction and then take the result to compute the product with the bra wavefunction to the left. In the practice of molecular spectroscopy l /) is commonly a collection, or set, of subwavefunctions l /,) whose subscript index i runs through the number n that is equal to the number of allowed static states of the molecule under study. Equation 7.6 also implies the Dirac function equality... [Pg.114]

Theoretically, the best possible input pulse would be an impulse or a Dirac function S, y. The Fourier transformation of is equal to unity at all frequencies. [Pg.515]

Fermi energy — The Fermi energy of a system is the energy at which the Fermi-Dirac distribution function equals one half. In metals the Fermi energy is the boundary between occupied and empty electronic states at absolute temperature T = 0. In the Fermi-Dirac statistics the so-called Fermi function, which describes the occupation fraction as a function of energy, is given by f(E) = —pjrj—> where E is the energy, ft is the - chem-... [Pg.269]

The Fourier transform of an infinite short pulse function h(t) = Kb(t), where 5(f) is Dirac s delta function, equals//(jco) = K, that is, it contains all the frequencies with the same amplitude K. Such a function caimot be realized in practice and must be substituted by a pulse of a short duration At. However, such a function does not have uniform response in the Fourier (i.e., frequency) space. The Fourier transform of such a function, defined as h(t) = 1 for r = 0 to To and h(t) = 0 elsewhere, equals... [Pg.163]

As A - 0 we take the function shown in Figure 7.3b. This function is called the unit impulse or Dirac function and it is usually represented by 5(t). It is defined as equal to zero for all times except for t = 0. Since the area under the unit pulse remains equal to 1, it is clear that this is true for the unit impulse ... [Pg.435]

In the case of an infinitely thin source - when the radioactive parent forms a monatomic layer on the front of the source at x = 0, and therefore p(x) = 5(x), where 5 is the Dirac function - lines 1 and 3 inO Eq. (25.29) are equal to (1 — fs) and 1, respectively. Therefore, one obtains the following formula, generally called the transmission integral ... [Pg.1394]

Fig. 44. - Left The dependence of the degree of disorder, Q on temperature, T, for (a) ideal limiting phase transitions of the l and 11" order, (b) diffuse phase transition of the and II " order and (c) an approximate function — the degree of assignment of ZO) ands Z(H). Right the approximation of phase transitions using two stepwise Dirac function designated as L and their thermal change dUdT. (A) Anomalous phase transitions with a stepwise change at the transformation point. To (where the exponent factor n equals unity and the multipfication constant has values -1, -2 and oo). (B) Diffuse phase transition with a continuous change at tlic point, T with the same multiplication constant but having the exponent factor 1/3. ... Fig. 44. - Left The dependence of the degree of disorder, Q on temperature, T, for (a) ideal limiting phase transitions of the l and 11" order, (b) diffuse phase transition of the and II " order and (c) an approximate function — the degree of assignment of ZO) ands Z(H). Right the approximation of phase transitions using two stepwise Dirac function designated as L and their thermal change dUdT. (A) Anomalous phase transitions with a stepwise change at the transformation point. To (where the exponent factor n equals unity and the multipfication constant has values -1, -2 and oo). (B) Diffuse phase transition with a continuous change at tlic point, T with the same multiplication constant but having the exponent factor 1/3. ...
Thus, the summation is equal to the Dirac delta function (see Appendix C)... [Pg.76]

When considering the composition inhomogeneity of Markovian copolymers, the finiteness of the chemical size of macromolecules cannot be ignored, because fractional composition distribution W(/ f) in the limit / -> oo turns out to be equal to the Dirac delta function 5(f - X). For macromolecules of finite size f2> 1 the function W(/ f) is the Gaussian distribution whose center and dispersion (Eq. 2) are described by relationships (Eq. 8) and the following one... [Pg.148]

The kernels of these integral equations, which are derived from simple probabilistic considerations, represent up to the factor 1 the product of two factors. The first of them, wa(r]), is equal to the fraction of a-th type blocks, whose lengths exceed rj. The second one, Vap(rj), is the rate with which an active center located on the end of a growing block of monomeric units M with length r) switches from a-th type to /i-lh type under the transition of this center from phase a into phase /3. The right-hand side of Eq. 74 comprises items equal to the product of the rate of initiation Ia of a-th type polymer chains and the Dirac delta function <5( ). [Pg.185]

Let us now define an infinite sequence of unit impulses or Dirac delta functions whose strengths are all equal to unity. One unit impulse occurs at every sampling time. We will call this series of unit impulses, shown in Fig. 18.4, the function /, . [Pg.620]

Molecules are more difficult to treat accurately than atoms, because of the reduced symmetry. An additional complication arises in relativistic calculations the Dirac-Fock-(-Breit) orbitals will in general be complex. One way to circumvent this difficulty is by the Douglas-Kroll-Hess transformation [57], which yields a one-component function with computational effort essentially equal to that of a nonrelativistic calculation. Spin-orbit interaction may then be added as a perturbation, implementation to AuH and Au2 has been reported [58]. Progress has also been made in the four-component formulation [59], and the MOLFDIR package [60] has been extended to include the CC method. Application to SnH4 has been described [61] here we present a recent calculation of several states of CdH and its ions [62], with one-, two-, and four-component methods. [Pg.170]

As one can see, for each set of parameters Z, k, L and d, one can select a in such a way that Eqs. (21) and (22) are fulfilled. Then, the resulting expectation values correspond to the variational minima and are equal to the appropriate exact eigenvalues of either Dirac or Schrodinger (or rather Levy-Leblond) Hamiltonian. However the corresponding functions do not fulfil the pertinent eigenvalue equations they are not eigenfunctions of these Hamiltonians. This example demonstrates that the value of the variational energy cannot be taken as a measure of the quality of the wavefunction, unless the appropriate relation between the components of the wavefunction is fulfilled [2]. [Pg.182]

Fig. 1. The ground state energies of a Z = 90 hydrogen-like atom obtainedfrom the Dirac (D) andfrom the Levy-Leblond (L) equations as functions of the nonlinear parameters. In the upper-rowfgures (Dl and LI) the a (abscissa) and (3 (ordinate) dependence ofE is displayed when L and S are set equal to the values corresponding to the exact solutions. In the lower-row figures (D2 and L2 ) the L (abscissa) and S (ordinate) dependence of E is displayed when a and 5 are set equal to the exact value. The arrows show directions of the gradient their length is proportional to the value of the gradient The solid line crossing the saddle corresponds to the functions andLl) or S = Sj (L) D2 and L2. For the definitions of thesefunctions see text. Fig. 1. The ground state energies of a Z = 90 hydrogen-like atom obtainedfrom the Dirac (D) andfrom the Levy-Leblond (L) equations as functions of the nonlinear parameters. In the upper-rowfgures (Dl and LI) the a (abscissa) and (3 (ordinate) dependence ofE is displayed when L and S are set equal to the values corresponding to the exact solutions. In the lower-row figures (D2 and L2 ) the L (abscissa) and S (ordinate) dependence of E is displayed when a and 5 are set equal to the exact value. The arrows show directions of the gradient their length is proportional to the value of the gradient The solid line crossing the saddle corresponds to the functions andLl) or S = Sj (L) D2 and L2. For the definitions of thesefunctions see text.
In this paper we present the first application of the ZORA (Zeroth Order Regular Approximation of the Dirac Fock equation) formalism in Ab Initio electronic structure calculations. The ZORA method, which has been tested previously in the context of Density Functional Theory, has been implemented in the GAMESS-UK package. As was shown earlier we can split off a scalar part from the two component ZORA Hamiltonian. In the present work only the one component part is considered. We introduce a separate internal basis to represent the extra matrix elements, needed for the ZORA corrections. This leads to different options for the computation of the Coulomb matrix in this internal basis. The performance of this Hamiltonian and the effect of the different Coulomb matrix alternatives is tested in calculations on the radon en xenon atoms and the AuH molecule. In the atomic cases we compare with numerical Dirac Fock and numerical ZORA methods and with non relativistic and full Dirac basis set calculations. It is shown that ZORA recovers the bulk of the relativistic effect and that ZORA and Dirac Fock perform equally well in medium size basis set calculations. For AuH we have calculated the equilibrium bond length with the non relativistic Hartree Fock and ZORA methods and compare with the Dirac Fock result and the experimental value. Again the ZORA and Dirac Fock errors are of the same order of magnitude. [Pg.251]

Fig. 5. The ground state energies of a Z 30 hydrogen-like atom in the Gaussian basis obtained from the Dirac (D) andfrom the L y-Leblond (L) equations as functions of a and. The saddle point coordinates equal to (282,168) in the Dirac case and (271,164) in the Levy-Leblondcase. The cross-sections of... Fig. 5. The ground state energies of a Z 30 hydrogen-like atom in the Gaussian basis obtained from the Dirac (D) andfrom the L y-Leblond (L) equations as functions of a and. The saddle point coordinates equal to (282,168) in the Dirac case and (271,164) in the Levy-Leblondcase. The cross-sections of...
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. [Pg.231]

The family of curves represented by eqn. (46) is shown in Fig. 11 and the mean and variance of both the E(f) and E(0) RTDs are as indicated in Table 5. When N assumes the value of 0, the model represents a system with complete bypassing, whilst with N equal to unity, the model reduces to a single CSTR. As N continues to increase, the spread of the E 0) curves reduces and the curve maxima, which occur when 0 = 1 —(1/N), move towards the mean value of unity. When N tends to infinity, E(0) is a dirac delta function at 0 = 1, this being the RTD of an ideal PER. The maximum value of E(0), the time at which it occurs, or any other appropriate curve property, enables the parameter N to be chosen so that the model adequately describes an experimental RTD which has been expressed in terms of dimensionless time see, for example. Sect. 66 of ref. 26 for appropriate relationships. [Pg.250]

This expression nicely illustrates the main qualitative features of the order (Za) nuclear size contribution. First, we observe a logarithmic enhancement connected with the singularity of the Dirac wave function at small distances. Due to the smallness of the nuclear size, the effective logarithm of the ratio of the atomic size and the nuclear size is a rather large number it is equal to about —10 for the IS level in hydrogen and deuterium. The result in (6.35) contains all state-dependent contributions of order (Za) . [Pg.123]

The usual uncertainty relations are a direct mathematical consequence of the nonlocal Fourier analysis therefore, because of this fact, they have necessarily nonlocal physical nature. In this picture, in order to have a particle with a well-defined velocity, it is necessary that the particle somehow occupy equally all space and time, meaning that the particle is potentially everywhere without beginning nor end. If, on the contrary, the particle is perfectly localized, all infinite harmonic plane waves interfere in such way that the interference is constructive in only one single region that is mathematically represented by a Dirac delta function. This implies that it is necessary to use all waves with velocities varying from minus infinity to plus infinity. Therefore it follows that a well-localized particle has all possible velocities. [Pg.537]

Thus resulting in the equality 1+ k. r ) = +Ao(k, r/). After a similar calculation for Bo and Co functions, the spectral component of Dirac s dyadic distribution is finally found to have the following expression ... [Pg.580]

Physicist P. A. M. Dirac suggested an inspired notation for the Hilbert space of quantum mechanics [essentially, the Euclidean space of (9.20a, b) for / — oo, which introduces some subtleties not required for the finite-dimensional thermodynamic geometry]. Dirac s notation applies equally well to matrix equations [such as (9.7)-(9.19)] and to differential equations [such as Schrodinger s equation] that relate operators (mathematical objects that change functions or vectors of the space) and wavefunctions in quantum theory. Dirac s notation shows explicitly that the disparate-looking matrix mechanical vs. wave mechanical representations of quantum theory are actually equivalent, by exhibiting them in unified symbols that are free of the extraneous details of a particular mathematical representation. Dirac s notation can also help us to recognize such commonality in alternative mathematical representations of equilibrium thermodynamics. [Pg.324]


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