Differentiability eigenfunctions

Before giving further motivations, we would like to recall the basic aspects concerned with the elementary problem of determining eigenfunctions and eigenvalues for the differential equation... 

In the next step, which is numerically the most demanding, the differential equations (3) are solved. Two possible strategies using a variational expansion of the single particle wave functions, /., are described below. After the eigenvalues and eigenfunctions have been found, a new ("output") charge density can be... 

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 ... 

The mean square segment displacements, which are the key ingredient for a calculation of the dynamic structure factor, are obtained from a calculation of the eigenfunctions of the differential Eq. 5.13. After retransformation from Fourier space to real space B k,t) is given by Eq. 41 of [213]. For short chains the integral over the mode variable q has to be replaced by the appropriate sum. Finally, for observation times mean square displacements can be expressed in... 

The differential virial theorem (169) for noninteracting systems can alternatively be obtained [31], [32] by summing (with the weights fj ) similar relations obtained for separate eigenfunctions 4>ja(r) of the one-electron Schrodinger equation (40) [in particular the KS equation (50)]. Just in that way one can obtain, from the one-electron HF equations (33), the differential virial theorem for the HF (approximate) solution of the GS problem, as is shown in Appendix B, Eq. (302), in a form ... 

The differential operator H describes the energy ohservahle in the sense that the eigenfunctions of this differential operator, i.e., wave functions (j)E satisfying H(/) = , with e R, arc the base states of the energy observable (see Assumption 3 of Section 1.2) and the probability of getting the result E from an energy measurement of an electron in the state is... 

The central idea for solving Eq. 1 can be explained without the details of the mathematical formalism. With an appropriate linear combination of the variables ya we can define new variables za (called eigenfunctions) such that the resulting differential equation is ... 

This operation very much simplifies the finding of eigenvalues and the corresponding eigenfunctions. To find them it is necessary to solve a differential equation in the following form ... 

In this example, we see that by differentiating the function y(x) = cos kx twice, we regenerate our original function multiplied by a constant which, in this case, is — k2. Hence, cos kx is an eigenfunction of A, and its eigenvalue is -k2. 

The procedure used to solve second-order differential equations of the form of equation (7.50) is essentially the same as that described in Worked Problem 7.4 and involves the construction of trial eigenfunctions from some of the functions introduced in Chapter 2. 

E.C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Qaredon, Oxford, Vol. 1,1946,1962 Vol. 2,1958. 

It is generally true that the normalized eigenfunctions of an Hermitian operator such as the Schrodinger Ti constitute a complete orthonormal set in the relevant Hilbert space. A completeness theorem is required in principle for each particular choice of v(r) and of boundary conditions. To exemplify such a proof, it is helpful to review classical Sturm-Liouville theory [74] as applied to a homogeneous differential equation of the form... 

Volumes have been written about the red herring known as Schrodinger s cat. Without science writers looking for sensation, it is difficult to see how such nonsense could ever become a topic for serious scientific discussion. Any linear differential equation has an infinity of solutions and a linear combination of any two of these is another solution. To describe situations of physical interest such an equation is correctly prepared by the specification of appropriate boundary conditions, which eliminate the bulk of all possible solutions as irrelevant. Schrodinger s equation is a linear differential equation of the Sturm-Liouville type. It has solutions, known as eigenfunctions, the sum total of which constitutes a state function or wave function, which carries... 

The second complication is that the equation, as traditionally interpreted, only handles point particles, but produces eigenfunction solutions of more complex geometrical structure. By analogy with electromagnetic theory the square of the amplitude function could be interpreted as matter intensity, but this is at variance with the point-particle assumption. The standard way out is to assume that ip2 represents a probability density rather than intensity. Historical records show that this interpretation of particle density was introduced to serve as a compromise between the rival matrix and differential operator theories of quantum observables, although eigenvalue equations, formulated in either matrix or differential formalism are known to be mathematically equivalent. 

In principle, knowledge of Eqs. [18]—[22] is sufficient to set up differential equations for the orbital angular momentum operators and to solve for eigenvalues and eigenfunctions. The solutions are most easily obtained employing spherical coordinates r, 0,< ) (see Figure 7). The solutions, called spherical harmonics, can be found in any introductory textbook of quantum chemistry and shall be given here only for the sake of clarity. 

Here we have suppressed the channel (q) index for convenience, assuming that it is contained in the n index. As usual, we insert Eq. (10.2) into the time-dependent Schrodinger equation, iWW/dt — H Vit), and use the orthogonality of the eigenfunctions of Hm, to obtain an indenumerable set of first-order differential equations that are analogous to Eq. (2.3) ... 

The basis behind separation of variables is the orthogonal expansion technique. The method of separation of variables produces a set of auxiliary differential equations. One of these auxiliary problems is called the eigenvalue problem with its eigenfunction solutions. 

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