The helium spectrum an empirical formula

The purpose of this section is to derive an analytical formula for the positions of bound states and resonances of the one-dimensional model. The derivation of the energy formula proceeds in two steps. First, we use a 

For a perturbative calculation of the spectrum of (10.4.1) we assume that electron 1 is in the single-particle SSE state m) and electron 2 is in a highly excited state, much more weakly bound than electron 1. In this situation electron 2 spends most of its time in regions with X2 xi. Therefore, we assume that the eigenfunctions of (10.4.1) are approximately of the form 

Neglecting v in (10.4.15), the mean field equation is solved by the zeroth order wave functions 

The energy shift AEmn in first order perturbation theory is given by the expectation value of v with the zeroth order wave functions (10.4.16). An exact calculation of the expectation value is diflBcult. But since we are only interested in the scaling of AEmn in fn and n for large m, n, we evaluate it semiclassically. Semiclassically, iprn is given by  

This is exactly the energy formula (10.4.9) used in Section 10.4.1. Here is of the form 1 + /(n/m), where the function /, a small correction, scales in the variable n/m. Since / is assumed to be small, we plotted — 1 versus n/m using all the data displayed in Fig. 10.8. We obtained a straight line, slightly bent at n/m 1, with an asymptotic slope of approximately —1.5 for n/m 1. This indicates that 1 approximately scales in the variable C + n/m. For C = 1 we obtain the result shown in Fig. 10.9. The data points, representing all the resonances computed in Section 10.4.1, approximately collapse on a straight line with slope w —1.5. Therefore we obtain an excellent fit to the data by 

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