Variation of wavefunctions with n and

We here anticipate that one consequence of the alkali model is the introduction of a minimum radius7 ro at which solutions of the Schrodinger equation for the inner and outer reaches of the alkali atom can be joined (this idea will be useful in chapter 3). 

The next point to note is the existence of so-called penetrating orbits,8 i.e. the occurrence of wavefunctions of the type shown in fig. 2.5. These are wavefunctions whose innermost loop penetrates the core, but which are otherwise quite hydrogenic, as expected in the outer reaches of the atom. For large enough n, such wavefunctions recapitulate as indeed do the radial wavefunctions of H this means that, apart from a normalisation factor, the inner part changes very slowly with increasing n beyond the first few values, as can be seen in fig. 2.5, and that the radial positions of all except the outermost nodes are stable. 

Recapitulation will be discussed in more detail in section 3.4. As n increases, the number of nodes also increases and, as n — oo, the inner nodes coincide with those of a continuum functions. In fact, the positions of the nodes determine the phase of the continuum function (which is oscillatory) at threshold. There is a simple relation between the phase shift above threshold and the quantum defect of the bound states, which will be explained in chapter 3. If the eigenfunctions recapitulate, i.e. the positions of the nodes are nearly constant, then it follows that the 

7 Note that ro denotes the radius of the core, and ao, the Bohr radius of H. 

8 This terminology is derived from the Bohr model of the atom, which was used in early descriptions of the alkali model, most notably by White [26]. 

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