Expansion in Terms of Eigenfunctions

An eigenfunction can usually be multiplied by a constant to normalize it, and we shall assume, unless stated otherwise, that all eigenfunctions are normalized  

As an example, consider the spherical harmonics. We shall prove that 

If Bf = kf where 1 is a constant, then use of the Hermitian property of B gives 

In the previous section, we proved the orthogonality of the eigenfunctions of a Hermitian operator. We now discuss another important property of these functions this property allows us to expand an arbitrary well-behaved function in terms of these eigenfunctions. 

We have used the Taylor-series expansion (Prob. 4.1) of a function as a linear combination of the nonnegative integral powers of (x - a). Can we expand a function as a linear combination of some other set of functions besides 1, (x - a), (jc - a), ... The answer is yes, as was first shown by Eourier in 1807. A Fourier series is an expansion of a function as a linear combination of an infinite number of sine and cosine functions. We shall not go into detail about Fourier series, but shall simply look at one example. 

Expansion of a Function Using Particle-in-a-Box Wave Functions. Let us consider expanding a function in terms of the particle-in-a-box stationary-state wave functions, which are [Eq. (2.23)] 

Before we can apply (7.29) to a specific f(x), we must derive an expression for the expansion coefficients a . We start by multiplying (7.29) by i/  

Now we integrate this equation from 0 to /. Assuming the validity of interchanging the integration and the infinite summation, we have 

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Friazinov (F4) deals with a generalized Stefan problem involving finite depth of the two-phase layer, densities and thermal conductivities which are functions of position, and arbitrary initial and boundary conditions, by an approximate expansion in terms of appropriate Sturm-Liouville eigenfunctions. 

Instead of proceeding with the coupled channel expansion of in terms of eigenfunctions of consider an adiabatic approximation to the y-motion. Thus, we shall express as... 

The formulas for the radiative and non-radiative decays rate in the quasi-static approximation were also derived by Gersten and Nitzan (GN) [48] who extended the quasi-static treatment to spheroids based on an expansion in terms of an orthogonal set of eigenfunctions, so that shape-induced shifts of radiative and non-radiative decay rates can be described. The accuracy of the GN decay rates versus the exact electrodynamic theory has been described in literature [47, 49, 50] for spherical nanoparticles, while no exact anal3d ic solution exists for spheroids [51]. 

Once our N-electron CSF has been reduced to a linear combination of one or two (N — 1)-electron spin eigenfunctions each multiplied by a creation operator, we may go one step further and expand each (N — l)-electron state in two (N — 2)-electron spin eigenfunctions as dictated by the penultimate element t -i in the genealogical coupling vector t. After N — I such steps, we arrive at an expansion in terms of determinants with projected spin M. In this way, we are led directly to an expansion of CSFs in Slater determinants where the coefficients are products of the genealogical coupling coefficients in (2.6.5) and (2.6.6). 

The statistical matrix may be written in the system of functions in which the coordinate x is diagonal. In one dimension, the eigenfunction of is the Dirac delta function. The expansion of (x) in terms of it is... 

Obviously, the state function (jc) is not an eigenfunction of H. Following the general procedure described above, we expand in terms of the eigenfunctions n). This expansion is the same as an expansion in a Fourier series, as described in Appendix B. As a shortcut we may use equations (A.39) and (A. 40) to obtain the identity... 

To resolve Eq. (19) an expansion of is made in terms of the zero-order eigenfunctions. Thus,... 

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 ... 

Thus, the expansion of / in terms of eigenstates of the property being measured dictated by the fifth postulate above is already accomplished. The only two terms in this expansion correspond to momenta along the y-axis of 2h/Ly and -2h/Ly the probabilities of observing these two momenta are given by the squares of the expansion coefficients of / in terms of the normalized eigenfunctions of -ihd/dy. The functions (1/Ly)l/2 exp(i27ty/Ly) and... 

The parameter is introduced to keep track of the order of the perturbation series, as will become clear. Indeed, one can perform a Taylor series expansion of the perturbed wave functions and perturbed energies using X to keep track of the order of the expansions. Since the set of eigenfunctions of the unperturbed SE form a complete and orthonormal set, the perturbed wave functions can be expanded in terms of them. Thus,... 

If the sj stem is small enough (a few light atoms) one can tackle the problem of solving exactly the QM coupled differential equations in the nuclear coordinates generated by the expansion of the global wavefunction of the system [23] in terms of the electronic eigenfunctions and the averaging over the electronic coordinates. 

The starting point of the present theory is an expansion of the time-dependent electronic wave function in terms of single-center eigenfunctions (pi of the target Hamiltonian jTte... 

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