Eigenvalue time-independent--------problem

Thus a wavepacket initiated in well A passes to well B by a curve crossing. Prof. Fleming showed an interesting case of persistent coherence in such a situation, despite the erratic pattern of the eigenvalue separations. An alternative, possibly more revealing approach, is to employ Stuckelberg-Landau-Zener theory, which relates the interference (i.e., coherence) in the two different wells via the area shown in Fig. 2. A variety of applications to time-independent problems may be found in the literature [1]. 

In Eq. (14), the position-dependent wavefunctions, y/ (k,r), and the eigenvalues, , are both obtained from the unperturbed, time-independent problem ... 

For the time dependent problem, we seek solutions of (2.1) for given initial conditions, and we are especially interested in the behavior of time independent problem, which is an eigenvalue problem, we seek to determine solutions of (2.6) corresponding to the largest (in modulus) eigenvalue A of (2.6). 

We remark that Theorem 2 assures the existence of a largest (in modulus) eigenvalue A of the time independent problem (3.2). Moreover, machine computations based on (3.10) are necessarily convergent, and Equation (3.13) of the Corollary gives for each iteration of (3.10) non-trivial upper and lower bounds on A, which is of considerable practical use, since A corresponds physically [17, p. 86] to the effective multiplication factor Kett- The bounds of (3.13) are also of numerical importance in determining when the iterative procedure of (3.10) has been carried far enough. ... 

We first assume that two states are available with equal energies. The eigenvalues of the time-independent problem are determined from the following secular equation ... 

Because of the spaiseness god the structure of the Hamiltonian matrix in a direct product DVR, solutions of both time dependent and time independent (eigenvalue) problems are made much more efficient compared with standard apmoaches. The two features exploited in the time dependent problems are the sparseness of H and the fact that the kinetic energy operators couple only one dimension at a time. This latter feature is exploited in the solution of time independent problems by sequential diagonalization and truncation in which "adiabatic" eigenvectors in lower dimensions are recoupled (exactly, within the basis) in the higher dimensions after truncation. We turn first to the time dependent problems. 

For a single electron, the time-independent Schrodinger eigenvalue problem is determined by the variational condition... 

In 32 the authors consider the solution of the two-dimensional time-independent Schrodinger equation by partial discretization. The discretized problem is treated as a problem of the numerical solution of system of ordinary differential equations and has been solved numerically by symplectic methods. The problem is then transformed into an algebraic eigenvalue problem involving real, symmetric, large sparse matrices. As numerical illustrations the authors have found the eigenvalues of the two-dimensional harmonic oscillator... 

In 34 the eigenvalue problem of the one-dimensional time-independent Schrodinger equation is studied. Exponentially fitted and trigonometrically fitted symplectic integrators are developed, by modification of the first and second order Yoshida symplectic methods. Numerical results are presented for the one-dimensional harmonic oscillator and Morse potential. 

A time-independent definition of such systems is contained in the characteristic spectrum of Lc. For example, denoting the eigenvalue associated with the eigenvalue problem Lcp = Ap as A, we have12... 

Since its eigenvalues correspond to the allowed energy states of a quantum-mechanical system, the time-independent Schrodinger equation plays an important role in the theoretical foundation of atomic and molecular spectroscopy. For cases of chemical interest, the equation is always easy to write down but impossible to solve exactly. Approximation techniques are needed for the application of quantum mechanics to atoms and molecules. The purpose of this subsection is to outline two distinct procedures—the variational principle and perturbation theory— that form the theoretical basis for most methods used to approximate solutions to the Schrodinger equation. Although some tangible connections are made with ideas of quantum chemistry and the independent-particle approximation, the presentation in the next two sections (and example problem) is intended to be entirely general so that the scope of applicability of these approaches is not underestimated by the reader. 

The continuous spectrum is also present, both in physical processes and in the quantum mechanical formalism, when an atomic (molecular) state is made to interact with an external electromagnetic field of appropriate frequency and strength. In conjunction with energy shifts, the normal processes involve ionization, or electron detachment, or molecular dissociation by absorption of one or more photons, or electron tunneling. Treated as stationary systems with time-independent atom - - field Hamiltonians, these problems are equivalent to the CESE scheme of a decaying state with a complex eigenvalue. For the treatment of the related MEPs, the implementation of the CESE approach has led to the state-specific, nonperturbative many-electron, many-photon (MEMP) theory [179-190] which was presented in Section 11. Its various applications include the ab initio calculation of properties from the interaction with electric and magnetic fields, of multiphoton above threshold ionization and detachment, of analysis of path interference in the ionization by di- and tri-chromatic ac-fields, of cross-sections for double electron photoionization and photodetachment, etc. 

The Variation Theorem. Given a system whose Hamiltonian operator H is time independent and whose lowest-energy eigenvalue is if is any normalized, well-behaved function of the coordinates of the system s particles that satisfies the boundary conditions of the problem, then... 

Evaluation of the energy in the time-independent Schrodinger equation requires the solution of an eigenvalue-eigenvector problem [22]. For an electronic wave function satisfying Eq. (2.9), an eigenvector— the total electronic energy— wiU be found. A possible poly electronic wave function for n electrons could have the form of a Hartree product ... 

Theorem 3. The time-independent eigenvalue problem with no up-scattering Equation (9), has a positive dominant eigenvalue Ao (i.e., Ao > Aa for all Aa Ao) which is simple corresponding to an eigenfunction O = ( J, ... 

Other methods for accelerating the convergence of the outer iterations exist, and are interesting in their own right. With the definition of the matrices in 3, we now write our discrete time independent eigenvalue problem in the matrix form... 

We are interested in eigenvalue problems because the time-independent Schrodinger equation is an eigenvalue equation ... 

The eigenvalues of this eigenvalue problem are found by diagonalizing the matrix M. The situation is more complicated if collisional relaxation is not fast compared to chemical reaction. In that case, the solution will yield more than one small negative eigenvalues and the overall reaction will proceed on a non-exponential time scale. In other words, if collisional relaxation interferes with unimolecular reaction, the reaction process cannot be described by a time-independent rate constant A uni-We now take a look at the corresponding chemically activated reaction,... 

Solution of the Lippmann-Schwinger-like equation in Brillouin-Wigner form, equation (55), for the reaction operator followed by solution of the eigenvalue problem (49) for the effective hamiltonian given in equation (52) is entirely equivalent to the solution of the time-independent Schrodinger equation, equation (1), for the state a. Furthermore, although recursion leads to the expansion (56), equation (55) remains valid when the series expansion does not converge. Equation (55) can be written... 

The above problem can be simplified by separating the fast electronic motion from the slow nuclear motion. One first defines an electronic Hamiltonian, also called clamped nucleus Hamiltonian Hgiir R) = Tgiir) + V r R). This electronic Hamiltonian acts in the electronic space and depends parametrically on the nuclear coordinates R, as indicated by the semicolon in the coordinate dependence of the operators. The eigenfunctions and eigenvalues of the associated time-independent Schrodinger equation (TISE)... 

In this section, we discuss briefly the generalized Floqnet formnlation of TDDFT [28,60-64]. It can be applied to the nonperturbative stndy of mnltiphoton processes of many-electron atoms and molecules in intense periodic or qnasi-periodic (multicolor) time-dependent fields, allowing the transformation of time-dependent Kohn-Sham equations to an equivalent time-independent generalized Floquet matrix eigenvalue problems. 

Schrddinger s time-dependent equation (6-1) is not an eigenvalue equation. However, Eq. (6-2) shows that Schrodinger s time-dependent equation is satisfied by time-independent eigenfunctions of H if they are multiplied by their time-dependent factors fit) = Q i—iEt/h). Furthermore, Eq. (6-2) continues to be satisfied if the term i if) fit) is replaced by a sum of such terms. (See Problem 6-9.) This means that we can seek to express the time-dependent state function, Pfx, 0. as a sum of time-independent box eigenfunctions as long as each of these is accompanied by its time factor fit). When i = 0, all the factors / it) equal unity, so at that point in time becomes the same as the sum of box eigenfunctions without their time factors. 

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