Eigenvalue problems eigenfunctions

In solving the eigenvalue problem for the energy operator //op, we have previously always introduced a complete basic set Wlt in which the eigenfunction has been expanded ... 

The search for eigenfunctions and eigenvalues in the example of the simplest difference problem. The method of separation of variables being involved in the apparatus of mathematical physics applies equelly well to difference problems. Employing this method enables one to split up an original problem with several independent variables into a series of more simpler problems with a smaller number of variables. As a rule, in this situation eigenvalue problems with respect to separate coordinates do arise. Difference problems can be solved in a quite similar manner. 

The eigenfunction of the vibrational ground state is calculated on the ab initio 2D So potential energy surface by solving the eigenvalue problem. 

By separation of variables, we immediately obtain the eigenvalues and eigenfunctions of the three-dimensional problem (A.49). 

The exact approach to the problem of dynamic (linear) stability is based on the solution of the equations for small perturbations, and finding eigenvalues and eigenfunctions of these equations. In a conservative system a variational principle may be derived, which determines the exact value of eigenfrequency... 

In order to separate the electronic and nuclear coordinates in an eigenvalue problem for the Hamiltonian defined by Equation 1, the adiabatic approximation in the version of a Bom-Oppenheimer model is used. In general, the eigenfunction defined within the adiabatic approximation is defined as a linear combination. 

We now show how the fundamental problem of quantum mechanics, the finding of the eigenvalues and eigenfunctions of F, G,... is formulated using matrix algebra. 

Because of the spin-spin coupling term, (8.41) is not separable into the sum of Hamiltonians for the individual nuclei, and the corresponding Schrodinger equation is not separable. To deal with the problem, we shall use the method of expanding the unknown wave functions in terms of a known complete set of functions. The eigenvalues and eigenfunctions (eigenvectors) are obtained as the solutions of (2.68) [or (2.77)] and (2.67). 

QUANTUM NUMBER. A number assigned to one of the various quantities that describe a particle or state. Many different characteristics of atomic and nuclear systems, as well as of those entities that arc introduced as a part of particle physics, are described by means of quantum numbers. The quantum numbers arise from the mathematics of the eigenvalue problem and may be related to the number of nodes in the eigenfunction. Any state may be described by giving a sufficient set of compatible quantum numbers, In the customary formulations, each quantum number is either an integer (which may be positive, negative, or zero) or an odd half-integer. 

Understand the basis of the eigenvalue problem and identify eigenfunctions, eigenvalues and operators... 

Which of (a)-(d) would be classified as eigenvalue problems What is the eigenfunction and what is the eigenvalue in each case ... 

Discussion of operators, the eigenvalue problem and associated eigenfunctions. 

Given that the total hamiltonian may be written as Hw = P2/2M + hw(R), the adiabatic eigenfunctions a R) are the solutions of the eigenvalue problem, hw(R) ot R) = Ea(R) a R). In this adiabatic basis the quantum-classical Liouville operator has matrix elements [12],... 

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. 

We note that the diffusion operator with Neumann (or periodic) boundary conditions is symmetric and has a simple zero eigenvalue with a constant eigenfunction. Equivalently, the eigenvalue problem... 

It thereby becomes an eigenvalue problem, since only a few values of W are eigenvalues for which corresponding eigenfunctions 9 exist which satisfy the equation and the boundary conditions. 

Unsal M., 1990, A solution for the complex eigenvalues and eigenfunctions of periodic Graetz problem, International Communications in Heat Mass Transfer, 25, 4, 585-592. 

The auxiliary eigenvalue problems of eqs. (2.a, b) md (6.a, b) are solved for tiie eigenvalues and related normalized eigenfunctions, either in analytic explicit form when apphcable or tiirough the GITT itself [35]. 

The boundary value problem posed by the differential equation (2.166) and the two boundary conditions (2.168) and (2.169) leads to the class of Sturm-Liouville eigenvalue problems for which a series of general theorems are valid. As we will soon show the solution function F only satisfies the boundary conditions with certain discrete values /q of the separation parameter. These special values /q are called eigenvalues of the boundary value problem, and the accompanying solution functions Fi are known as eigenfunctions. The most important rules from the theory of Sturm-Liouville eigenvalue problems are, cf. e.g. K. Janich [2.33] ... 

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