Bound-state energies

The energies of the selective adsorption resonances are very sensitive to the details of the physisorption potential. Accurate measurement allied to computation of bound state energies can be used to obtain a very accurate quantitative fonn for the physisorption potential, as has been demonstrated for helium atom scattering. For molecules, we have... 

Gesztesy, F Grosse, H., and Thaller, B., 1984, A rigorous approach to relativistic corrections of bound state energies for spin-1/2 particles , Ann. Inst. Poincare 40 159. 

Orthogonal Polynomial Expansion of the Spectral Density Operator and the Calculation of Bound State Energies and Eigenfunctions. 

Time-to-Energy Fourier Resolution Method for Calculating Bound State Energies and Wavefunctions. 

To evaluate the linear term of the bound state energy shift within perturbation theory we need the bound state wave function. For this, we have to specify the interaction. We will adopt the following parameterization of the nucleon-nucleon interaction. 

For our exploratory calculation we use Guaussian type wave functions to find optimal bound states in the three and four particle case. Then we are able to calculate the perturbative expression for the shift of the bound states energy... 

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... 

The y>Ee(R) are the radial free-state wavefunctions (see Chapter 5 for details). The free state energies E are positive and the bound state energies E(v,S) are negative v and ( are vibrational and rotational dimer quantum numbers t is also the angular momentum quantum number of the fth partial wave. The g( are nuclear weights. We will occasionally refer to a third partition sum, that of pre-dissociating (sometimes called metastable ) dimer states,... 

We expect to present applications of these formulae shortly, but some of the present results are perhaps interesting as such, e. g. the fact that in an arbitrary system the bound state energies decrease when the "strength" of the potential increases. 

Model calculations (e.g. [4]) predict that TqD exhibits resonances at the bound state energies of the QD, while aQp exhibits an interesting variation... 

Variational calculation of bound-state energies and wavefunctions... 

In order to calculate bound-state energies and the corresponding eigenfunctions we expand I>(R,r) according to... 

The calculation of bound-state energies is part of molecular spectroscopy. Many efficient methods and computer codes have been developed... 

The theory outlined above can be used to calculate the exact bound-state energies and wavefunctions for any triatomic molecule and for any value J of the total angular momentum quantum number. We can solve the set of coupled equations (11.7) subject to the boundary conditions Xjfi (R Jp) —> 0 in the limits R —> 0 and R — oo (Shapiro and Balint-Kurti 1979). Alternatively we may expand the radial wavefunctions in a suitable set of one-dimensional oscillator wavefunctions ipm(R),... 

In order to calculate the bound-state energies one expands the two-dimensional wavefunction in terms of products of suitably chosen, onedimensional basis functions (fn(Ri), = 1,2, which describe the vibration of the two OH entities within H20(X). Because the Hamiltonian is symmetric with respect to the interchange of the two bond distances, the... 

In order to obtain a realization of so(4), the factor — 2H must be removed from Eq. (162c). There are two possibilities H can be replaced by one of its continuum or bound-state energy eigenvalues. The former choice leads to a realization of so 3, 1) and the latter to a realization of, so(4). Thus, if we replace H by the bound-state energy E , and define the modified Laplace-Runge-Lenz vector... 

It is interesting to note that straightforward Bohr-Sommerfeld quantization of the action (6.1.11) yields the exact result (6.1.25) for the bound state energies. In our units the Bohr-Sommerfeld condition results in / = n, n = 1,2,. Inserting this result into (6.1.13) indeed reproduces (6.1.25) exactly. This is the same happy accident which allowed Bohr (1913) to obtain the Balmer formula from a simple solar system model of a one-electron atom. 

The method defined in (6.2.58) turns the real bound state energies into complex resonances, where the width of the resonances, A /2, is assumed... 

For calculating bound-state energies and wave functions, A 0 and the filter becomes equivalent to the spectral density operator S Ei — H) [208]. In the case of resonances, the Green s function in Eq. (33) selects the contributions from those complex poles whose real parts lie near Ei. In what follows we consider only the filter defined in Eq. (33). 

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