Eigenfunction period

For bound state systems, eigenfunctions of the nuclear Hamiltonian can be found by diagonalization of the Hamiltonian matiix in Eq. (11). These functions are the possible nuclear states of the system, that is, the vibrational states. If these states are used as a basis set, the wave function after excitation is a superposition of these vibrational states, with expansion coefficients given by the Frank-Condon overlaps. In this picture, the dynamics in Figure 4 can be described by the time evolution of these expansion coefficients, a simple phase factor. The periodic motion in coordinate space is thus related to a discrete spectrum in energy space. 

The quantization ofLz arises because the eigenfunctions fnust be periodic in 0 ... 

It may serve to make these results more transparent if we write them out in terms of the familiar momentum eigenfunctions of a particle in a finite box of side L with periodic boundary conditions.12 In this case we have... 

One way in which we can solve the problem of propagating the wave function forward in time in the presence of the laser field is to utilize the above knowledge. In order to solve the time-dependent Schrodinger equation, we normally divide the time period into small time intervals. Within each of these intervals we assume that the electric field and the time-dependent interaction potential is constant. The matrix elements of the interaction potential in the basis of the zeroth-order eigenfunctions y i Vij = (t t T(e(t)) / ) are then evaluated and we can use an eigenvector routine to compute the eigenvectors, = S) ... 

The term scar was introduced by Heller in his seminal paper (Heller, 1984), to describe the localization of quantum probability density of certain individual eigenfunctions of classical chaotic systems along unstable periodic orbits (PO), and he constructed a theory of scars based on wave packet propagation (Heller, 1991). Another important contribution to this theory is due to Bogomolny (Bogomolny, 1988), who derived an explicit expression for the smoothed probability density over small ranges of space and energy... 

Certain semiclassical properties involving the eigenfunctions can also be calculated with periodic-orbit theory. Considering the Wigner functions corresponding to die energy eigenfunctions H = Enn [28],... 

The concentration of the Wigner transforms of the eigenfunctions on the quantized tori provides the counterpart in classically integrable systems to the scarring phenomena associated with unstable periodic orbits. 

Differences between the lifetimes obtained from equilibrium point quantization and periodic-orbit quantization appear as the bifurcation is approached. The lifetimes are underestimated by equilibrium point quantization but overestimated by periodic-orbit quantization. The reason for the upward deviation in the case of periodic-orbit quantization is that the Lyapunov exponent vanishes as the bifurcation is approached. The quantum eigenfunctions, however, are not characterized by the local linearized dynamics but extend over larger distances that are of more unstable character. 

The eigenfunctions associated with the resonances have been obtained via wavepacket propagation. They appear to be localized along the symmetric-stretch periodic orbit 0, with a number of nodes equal to n and even under the exchange of iodine nuclei. Due to the relative stability of the symmetric-stretch orbit, we have thus here a system where the hypothesis of the orbit 12 representing the RPO, that is, resonant periodic orbit, does not hold. 

Thus, for example, if we apply equation (7.46) to describe the periodic vibrational motion in a diatomic molecule, x represents time, and positive and negative values for y correspond to bond extension and compression, respectively. Finally, we can see that equation (7.46) is an eigenvalue equation in which y is the eigenfunction and —n2 is the eigenvalue (see Section 4.3.1). 

The chemical reaction corresponds to a preparation-registration type of process. With the volume periodic boundary conditions for the momentum eigenfunction, the set of stationary wavefiinctions form a Hilbert space for a system of n-electrons and m-nuclei. All states can be said to exist in the sense that, given the appropriate energy E, if they can be populated, they will be. Observe that the spectra contains all states of the supermolecule besides the colliding subsets. The initial conditions define the reactants, e.g. 1R(P) >. The problem boils down to solving eq.(19) under the boundary conditions defining the characteristics of the experiment. 

Founargiotakis, M., Farantos, S.C., Contopoulos, G., and Polymilis, C. (1989). Periodic orbits, bifurcations, and quantum mechanical eigenfunctions and spectra, J. Chem. Phys. 91, 1389-1401. 

Heller, E.J. (1984). Bound-state eigenfunctions of classically chaotic Hamiltonian systems Scars of periodic orbits, Phys. Rev. Lett. 53, 1515-1518. 

In the case of periodic boundary conditions the chain Hamiltonian commutes with the operator that displaces all electrons by one unit cell cyclically. Therefore, its eigenfunctions must be characterized by the hole quasi-... 

Molecular orbitals (MOs) were constructed using linear combinations of basis functions of atomic orbitals. The MO eigenfunctions were obtained by solving the Schrodinger equations in numerical form, including Is— (n+l)p, that is to say, Is, 2s, 2p, -ns, np, nd, (n+l)s, (n+l)p orbitals for elements from n-th row in the periodic table and ls-2p orbitals for O, where n—1 corresponded to the principal quantum number of the valence shell. 

In one dimension, the Bloch result reduces to an earlier Floquet55 result [22] In one dimension the periodicity of the lattice requires that the eigenfunction of the appropriate Hamiltonian must satisfy... 

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... 

The boundary conditions are periodic and the number of allowed values of the wave vector k is equal to the number of unit cells in the crystal. These eigenfunctions constitute the basis for the infinite-dimensional Hilbert space of the crystal Hamiltonian and any function with the same boundary conditions can be expressed as a linear combination of functions in this complete set. 

Mathematica package is developed that computes the eigenvalues, the eigenfunctions, the eigenintegrals, the dimensionless temperature, the average dimensionless temperature, and the Nusselt number for steady state and periodic heat transfer in micro parallel plate channel and micro tube taking into account the velocity slip and the temperature jump. Some results in form of tables and plots are given bellow. 

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. 

Table 1 Convergence behavior of eigenfunction expansion for the dimensionless velocity and comparison with routine NDSolve [41] (parallel plates, periodic flow, j5 =0.1).

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