The Bound-States Problem

3 The Bound-States Problem. - For negative energies we solve the so-called bound-states problem, i.e. with the boundary conditions 

For the purpose of obtaining our numerical results it is appropriate to choose v in the way suggested by Ixaru and Rizea6 and given in Section 2.4.1. 

In order to solve this problem numerically we use a strategy which has been proposed by Cooley and has been improved by Blatt. This strategy involves integrating forward from the point x = 0, backward from the point x = 15 and matching up the solution at some internal point in the range of integration. As initial conditions for the backward integration we take (see Cash et al )  

We have also applied the above methods to other potentials like the parabolic potential  

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Leading recoil corrections in Za (of order (Za) (m/M)") still may be taken into account with the help of the effective Dirac equation in the external field since these corrections are induced by the one-photon exchange. This is impossible for the higher order recoil terms which reflect the truly relativistic two-body nature of the bound state problem. Technically, respective contributions are induced by the Bethe-Salpeter kernels with at least two-photon exchanges and the whole machinery of relativistic QFT is necessary for their calculation. Calculation of the recoil corrections is simplified by the absence of ultraviolet divergences, connected with the purely radiative loops. 

This new integration region from the electron to the proton mass, which was discovered in [5], arises here for the first time in the bound state problem. As we will see below, especially in discussion of the hyperfine splitting, these high momenta are responsible for a number of important contributions to the energy shifts. 

The method of direct analysis of the integration regions applied to the bound state problem in [5, 6] allowed these authors also to obtain quadratic in mass ratio radiative-recoil corrections of order a Za) ... 

Ill) Bound-States Problem For the bound-states problem the most accurate methods are the methods derived by Simos and Williams (Case II, Case III, Case IV, Case V and Case VI of the family),20 the method derived by Simos,24... 

In this context, the idea of discrete numerical basis sets, introduced by Sa-lomonson and Oster (129) for the bound-state problem and combined with the complex-rotation method by Lindroth (30), is very interesting. One-particle basis functions are defined on a discrete grid inside a spherical box containing the system under cosideration. The functions are evaluated by diagonalizing the discretized one-particle complex-rotated Hamiltonian. Such basis sets are then used to compute autoionizing state parameters by means of bound-state methods (30,31,66). 

Suppose we want to treat quarkonium using QCD. How do we go about it Bound state problems cannot be treated by conventional perturbation theory. And so far, field theories can only be evaluated in perturbation theory. It seems as if there is a mismatch between field theory and the bound state problem. 

Therefore, it is not clear that lowest-order QCD perturbation theory is at all relevant to the bound-state problem. 

It is possible to formulate the bound state problem for a wide range of Van der Waals complexes in a form reminiscent of that for atom-diatom systems. The similarities between the different sets of coupled equations will be helpful in understanding the dynamical approximations that may be applied to the larger complexes. In particular, the helicity decoupling approximation, which has proved to be accurate for most atom-diatom systems, is equally applicable to the larger systems. 

R = 00 rather than just at = 0 as in the scattering case. These boundary conditions are in fact identical to those of closed channels in scattering theory. We now utilize this fact and set up an artificial scattering problem in which the bound-state problem of interest forms the closed channels of a scattering problem [264]. This in turn enables us to use with a modification the standard techniques of molecular scattering theory [276-279]. 

Mead and Truhlar [10] broke new ground by showing how geometric phase effects can be systematically accommodated in scattering as well as bound state problems. The assumptions are that the adiabatic Hamiltonian is real and that there is a single isolated degeneracy hence the eigenstates n(q-, Q) of Eq. (83) may be taken in the form... 

On pages 133-136 of Pauling and Wilson are tabulated the bound-state solutions of this problem for n ranging up to n=6 and 1-values up to 1=5. 

The method of superposition of configurations is essentially based on the assumption that the basic orbitals form a complete set. The most popular basis used so far in the literature is certainly formed by the hydrogen-like functions, which set contains a discrete and a continuous part. The discrete subset corresponds physically to the bound states of an electron around a proton, whereas the continuous part corresponds to a free electron scattered by a proton, or classically to the elliptic and hyperbolic orbits, respectively, in a central-field problem. 

In order to overcome the problems that can occur in sampling configurations of the unbound ligands when using a multiple topology model, we consider two types of restraining potentials. For simplicity, both restraining potentials disappear at the bound states... 

In recent years, state-of-the-art recursive diagonalization methods have been applied to bound-states problems for LiCN,152 H20,12,117,239-241 CH2,242 HCN,i3,80,105-107,241 40,67,164,243-245 246 H0C1,247,248 N02,76,249-253... 

If a ligand is available in isotope-labeled form, then the use of standard isotope-edited techniques (preferentially with at least one heteronuclear dimension) will allow straightforward access to its structure in the bound state without having to solve the much more complex problem of the protein structure. 

Basis sets of the type discussed in this paper can only be applied to bound-state problems. It is interesting to ask whether it might be possible to constmct many-electron Sturmian basis sets appropriate for problems in reactive scattering in an analogous way, using hydrogenlike continuum functions as building-blocks. We hope to explore this question in future publications. 

Contributions to the energy which depend only on the small parameters a. and Za. are called radiative corrections. Powers of a arise only from the quantum electrodynamics loops, and all associated corrections have a quantum field theory nature. Radiative corrections do not depend on the recoil factor m/M and thus may be calculated in the framework of QED for a bound electron in an external field. In respective calculations one deals only with the complications connected with the presence of quantized fields, but the two-particle nature of the bound state and all problems connected with the description of the bound states in relativistic quantum field theory still may be ignored. 

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