Comparison with ab initio theory recapitulation

As will be shown later, the tangent function is a more fundamental quantity, and so this equation is also written in the equivalent form 

The form of the wavefunction deduced by QDT is rather interesting. It differs from that of H, as we have seen, by the fact that the irregular Coulomb function g v,r) is necessarily present in the solution. Mathematically, this means that the zeros or nodes in rp occur at values which no longer depend solely on n as in the case of H, but on both n and p. At large enough n, the wavefunctions of H have the interesting property that their inner nodes recapitulate, i.e. that the radii at which all except the outermost nodes occur for successive values of n become stable as the series limit is approached. 

This property is actually not surprising as the series limit is approached, the bound state wavefunction acquires more and more nodes, and tends to the oscillatory function of the continuum. The position of the nodes is related to the phase in the continuum, and we may expect that the two are connected, since the wavefunction at very high n must change smoothly into the free electron s wavefunction just above the series limit. In QDT, as for H, the wavefunction for r ro preserves this 

1 Actually, QDT alone does not determine whether n should be positive or not. Indeed it says nothing at all about the correct value of n, a point which recurs repeatedly in applications of QDT to real data. 

The shift between the position of the nodes for H and for a general QDT wavefunction is controlled entirely by p and may be considered as a phase shift at high enough n. 

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