n-dependence of the energy values

The method in which the Coulomb potential is replaced by a suitably adapted box potential yields the correct $n$ -dependence of the hydrogen energy values: $E_{ges} \sim -1/n^2$ .

This raises a question: the energy values of an (ordinary) potential well are proportional to $n^2$ (cf. Equation $(2)$ in the main text). How, then, can a potential-well model yield an energy dependence of $1/n^2$ ?

Answer: the parameters of the potential well ( $R$ and $W_0$ ) themselves contain the energy $E_{ges}$ (cf. Equation $(1)$ in the main text). If these energy-dependent parameters are substituted into the energy formula $(2)$ , $E_{ges}$ appears on both sides of the equation. Solving the equation for $E_{ges}$ can then yield a different $n$ -dependence than the original $E \sim n^2$ — and this is exactly what happens here.

This aspect of the method presented is somewhat difficult to see through: in essence, a different potential well is fitted to each value of $E_{ges}$ .

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