Schrodinger eigenfunctions

The corrections of order (Za) are just the first order matrix elements of the Breit interaction between the Coulomb-Schrodinger eigenfunctions of the Coulomb Hamiltonian Hq in (3.1). The mass dependence of the Breit interaction is known exactly, and the same is true for its matrix elements. These matrix elements and, hence, the exact mass dependence of the contributions to the energy levels of order (Za), beyond the reduced mass, were first obtained a long time ago [2]... 

But he was not at all certain that the principles considered so far in atomic theory could, in fact, be used for the realization of such a program. This was because the characteristic interaction of the chemical forces deviated completely from other familiar forces These forces seemed to "awake" after a previous "activation," and they suddenly vanished after the "exhaustion" of the available "valences." By making use of elementary symmetry considerations, it was known that the mode of operation of the homopolar valence forces could be mapped onto the symmetry properties of the Schrodinger eigenfunction of the atoms of the periodic system and could be interpreted as quantum mechanical resonance effects. This interpretation was formally equivalent to its chemical model, that is, it produced the same valence numbers and... 

Prom here on it makes sense to consider systems which are separable in some point coordinate, so that the Schrodinger eigenfunctions can be written as products of functions which depend on only one variable each. If the system remains separable under a change of coordinates, then the following theorem is valid the energy is a monotonous function of each of the quantum numbers, when the quantum numbers are interpreted as the number of zeros of the factors of the eigenfunctions which depend on only one coordinate. FVom this it follows immediately that terms of which the separate quantum numbers differ by 1 do not cross for the adiabatic changes considered here. 

The use of c instead of a is said to constitute the interaction representation it is designed to remove H from the Schrodinger equation for c. Passage from a to c is equivalent to a change from ua to e immug, and in the special case where ua is an eigenfunction of H, the latter may be written e liln)B< tua. Hence, all matrices with components PIm(it) in the basis are transformed to P, m by the rule... 

The Time Reversal Operator.—In this section we show that spatial operators are linear whereas the time reversal operator is antilinear.5 This may be seen by examining the eigenfunctions of the time dependent Schrodinger equation... 

Perhaps the most straightforward method of solving the time-dependent Schrodinger equation and of propagating the wave function forward in time is to expand the wave function in the set of eigenfunctions of the unperturbed Hamiltonian [41], Hq, which is the Hamiltonian in the absence of the interaction with the laser field. 

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) ... 

PES), which is different for each electronic state of the system (i.e. each eigenfunction of the BO Schrodinger equation). Based on these PESs, the nuclear Schrodinger equation is solved to define, for example, the possible nuclear vibrational levels. This approach will be used below in the description of the nuclear inelastic scattering (NIS) method. 

As a simple application of the variation method to determine the ground-state energy, we consider a particle in a one-dimensional box. The Schrodinger equation for this system and its exact solution are presented in Section 2.5. The ground-state eigenfunction is shown in Figure 2.2 and is observed to have no nodes and to vanish at x = 0 and x = a. As a trial function 0 we select 0 = x(a — x), 0 X a... 

The mathematical procedure that we present here for solving equation (9.15) is known as Rayleigh-Schrodinger perturbation theory. There are other procedures, but they are seldom used. In the Rayleigh-Schrodinger method, the eigenfunctions tpn and the eigenvalues E are expanded as power series in A... 

The TD wavefunction satisfying the Schrodinger equation ih d/dt) F(f) = // (/,) can be expanded in a basis set whose elements are the product of the translational basis of R, vibrational wavefunctions for r, r2, and the body-fixed (BF) total angular momentum eigenfunctions as41... 

In a diabatic representation, the electronic wave functions are no longer eigenfunctions of the electronic Hamiltonian. The aim is instead that the functions are so chosen that the (nonlocal) non-adiabatic coupling operator matrix, A in Eq. (52), vanishes, and the couplings are represented by (local) potential operators. The nuclear Schrodinger equation is then written... 

Since the Hamiltonian is symmetric in space coordinates the time-dependent Schrodinger equation prevents a system of identical particles in a symmetric state from passing into an anti-symmetric state. The symmetry character of the eigenfunctions therefore is a property of the particles themselves. Only one eigenfunction corresponds to each eigenfunction and hence there is no exchange degeneracy. 

The formal similarity between Eq. (10) and the time-dependent Schrodinger equation is striking, and we shall indeed develop methods which are very reminiscent of quantum mechanics. In particular, we may calculate the eigenfunctions and eigenvalues of the unperturbed Liouville operator L0. We look for solutions of ... 

The bound-state energies and eigenfunctions can be obtained by solving the Schrodinger equation with boundary conditions that the radial wave function vanishes at both ends... 

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