Expansion theorem

The Hermitian property of quantum-mechanical operators leads to an important result. Consider two eigenfunctions, xj/ and i/, of the Hamiltonian operator we have 

We take the complex conjugate of the second equation, then we multiply 

the Hermitian property, the two integrals on the right are equal hence 

Equation (20.37) is the orthogonality relation. Two eigenfunctions of a linear Hermitian operator corresponding to distinct eigenvalues are orthogonal. Note that if fc = n, our normalization requirement is 

This concept of orthogonality can be obtained by extension from the concept of orthogonality of two ordinary vectors in three-dimensional space. If the x, y, and z components are a, Uy, for the first vector and b, by, b for the second vector, then the condition for orthogonality is 

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The validity of such an expansion theorem has been carefully studied in mathematics, and there is no need for going into details here. 

For many purposes, it is more convenient to express all functions with respect to just one origin - most usually the metal. The expansion theorem may be exploited to express any function as an (infinite) sum of convenient basis functions. Here we write the function centred on the ligand as a linear combination of functions centred on the metal... 

The linearisation of the non-linear component and energy balance equations, based on the use of Taylor s expansion theorem, leads to two, simultaneous, first-order, linear differential equations with constant coefficients of the form... 

By using the expansion theorem, the total wave function can be expanded in the noninteracting basis set ... 

We shall start with the definition of density matrix [82-84]. For this purpose, we consider a two-state system. According to the expansion theorem we have... 

We will apply these to a small control volume of radius b and height dz, as shown in Figure 10.5, and we will employ the Taylor expansion theorem to represent properties at z + dz to those at z ... 

Some care has to be exercised when demonstrating an expansion theorem in terms of Eq. (A.58), because the differential operator (A.52) is not Hermitian. It is, however, very easy to find a conjugate system of eigenfunctions 24 they are obtained by substituting —km for kx in Eq. (A.58). We then have for an arbitrary function ... 

Because of these orthogonality relationships wc can establish an expansion theorem which is a straightforward generalization of the Legendre series (16.3). It is readily shown that for a large class of functions /, the function /(0, [Pg.81]

The 1- and 2-RDMs can be obtained in the coordinate-space representation via the expansion theorem ... 

Generally, the atomic, outer, and interatomic sphere radii are chosen so as to minimize the volume between the spheres. Other physical considerations, such as empirical atomic or ionic radii, can also influence the initial choice of sphere sizes. It has also been recently shown that overlapping spheres may be used since the various expansion theorems are still valid (192). This technique has been used to calculate the molecular orbitals of Zeise s salt and ferrocene discussed later in this review (191, 193). 

We note that similar selection rules have been derived on the basis of determinantal product states, using the expansion theorem of Laplace [15]. The relationship between both formalisms is still under study [16],... 

The ket-bra operators are useful in many connections. Using (1.15), the expansion theorem (1.6) may now be written in the form... 

This is the general bi-orthogonality theorem valid for a pair of adjoint operators. From the expansion theorem x = Xk Ck ak, one gets further [Di x] = Ek [Di CJak = [Di Ci]ai. Since the product [Di x] cannot be... 

The connection between the two Hilbert spaces and L2 used in physics is established by using the existence of at least one orthonormal basis (p = (pk(X) in L2, which leads to the expansion theorem... 

The same result follows when the average of the exponential function given in Eq. (51) is transformed using the cumulant expansion theorem and, assuming a Gaussian process, all correlations higher than second order (Stepisnik, 1981, 1985) are neglected. The particle velocity autocorrelations form a tensor... 

In the next section we shall present a simplified expansion theorem of osmotic pressure which was first obtained by McMillan and Mayer. This cluster expansion theory will be further extended in Section 3 to distribution functions, and medn results of Kirkwood and Buff will be recovered. A new and simple derivation of the cluster expansion of the pair distribution function is also given. Section 4 presents a new expression for the chemical potential of solvents in dilute solutions. Section 5 shows how the general solution theory may be applied to compact macromolecules. Finally, Section 6 deals with the second osmotic virial coefficient of flexible macromolecules and is followaJ in Sa tion 7 by concluding remarks. 

The inverse Laplace transform of the matrix elements in equation (46b) can be found from Laplace transform tables or by using the Heaveside expansion theorem. For example, consider the entry in the rtjj position of equation (46b). We find from Table 8.1 of Varma and Morbidelli[3] that... 

In examples 8.3 and 8.4 Maple was used to invert from the Laplace domain to the time domain. Unfortunately, these two examples are very simple and, hence, we could invert to the time domain using Maple. For practical problems, inversion is not straightforward. The inversion to the time domain can be done in two different ways. In section 8.1.4, short time solutions will be obtained by converting the solution in Laplace domain to an infinite series. In section 8.1.5, a long time solution will be obtained by using the Heaviside expansion theorem. 

The short time solutions obtained in section 8.1.4 (examples 8.1.5 and 8.1.6) require only a few terms in the infinite series at short times to converge. However, at long times the series requires a large number of terms and cannot be used efficiently. The long time solution can be obtained using Heaviside expansion theorem.[l] If we denote the solution obtained in the Laplace domain as F(s) ... 

Differential Equations - Heaviside Expansion Theorem for Multiple Roots... 

Often inversion to time domain solution is not trivial and the time domain involves an infinite series. In section 8.1.4 short time solution for parabolic partial differential equations was obtained by converting the solution obtained in the Laplace domain to an infinite series, in which each term can easily inverted to time domain. This short time solution is very useful for predicting the behavior at short time and medium times. For long times, a long term solution was obtained in section 8.1.5 using Heaviside expansion theorem. This solution is analogous to the separation of variables solution obtained in chapter 7. In section 8.1.6, the Heaviside expansion theorem was used for parabolic partial differential equations in which the solution obtained has multiple roots. In section 8.1.7, the Laplace transform technique was extended to parabolic partial differential equations in cylindrical coordinates. In section 8.1.8, the convolution theorem was used to solve the linear parabolic partial differential equations with complicated time dependent boundary conditions. For time dependent boundary conditions the Laplace transform technique was shown to be advantageous compared to the separation of variables technique. A total of fifteen examples were presented in this chapter. 

There is a well known expansion theorem to expand the plane wave solution (which are the solutions to the free particle equation) in terms of Bessel functions as well [39] ... 

To solve this we make use of the expansion theorem discussed in the last section. We consider that the unknown functions n can be expanded in terms of the known functions since the latter form a normalized orthogonal set, and write... 

A wave function satisfying this equation is a function of the time and of the coordinates of the system. For a given value of t, say t, f it ) is a function of the coordinates alone. By the general expansion theorem of Section 22 it can be represented as a series involving the complete set of orthogonal wave functions for the unperturbed system,... 

The real version of the irreducible tensor method, related to the complex representations as mentioned, is highly useful in the ligand-field theory, as will be shown in the second part of this paper. An additional reason for this is the fact that expansion theorems concerning functions and operators achieve apt forms when the tensor method is applied to real spherical harmonics. 

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