Single channel QDT

This approach shows that the quantum defect results from the short range part of the potential. It provides parametrised wavefunctions associated with the jue, and suggests a more versatile method of handling the atomic core, described below. 

At a general energy e in atomic units, measured relative to the ionisation threshold, or in terms of the reduced energy variable v of QDT, defined by e = E00 — /2v2 (note that e is negative for bound states), the one-electron Schrodinger equation outside ro is just the same as for H. Thus, the solution involves two functions, f(u,r) and g(u,r), whose behaviour at the origin is different. We have 

So far, we have not specified the energy, and so v is not yet fixed at any specific value, but is a running variable. In order to quantise the energy, we must as usual impose boundary conditions appropriate for a bound state. These will turn out to quantise u, as we now demonstrate. 

In these expressions, u(v, r) increases exponentially with increasing r, and its coefficient in the final expression for ip must therefore vanish at large r, whereas v(v, r) tends to zero at large r and may therefore have a nonzero coefficient. Substituting into equation (3.4), we have 

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The organisation of the present chapter is as follows. First, we give the basic principles of single channel QDT. Next, we move on to series perturbations, and to simple graphical methods which can be used to analyse them by MQDT. Finally, we give examples of how such methods can be extended to more complex situations, and we deal with extrapolations beyond the series limit which allow autoionisation profiles to be predicted from the data for interacting bound states. 

The theory described in section 3.3 is called single channel QDT. It reduces an unperturbed Rydberg series to a small number of constants, which include /z. Were this the only achievement of QDT, it would not have assumed much importance in atomic physics, because most series in the spectra of many-electron atoms are in fact perturbed, so that a constant value of p is a rarity. 

Plots of quantum defects and of C(E) against energy are shown in fig. 5.14 (a) and (b). They demonstrate how this equation linearises QDT for a single channel containing a giant resonance. 

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