The form of the potential of the hydrogen atom is of course special, because it originates from Coulomb's law of electric force through canonical quantization. However, we can imagine a general form of a two-body, central potential like

\[\begin{equation*}V(\hat{r})=-\frac{k}{\hat{r}^{n}} \tag{9.173}\end{equation*}\]

for some \(n>0\), and constant \(k\) that has units of energy \(\times\) distance \({ }^{n}\). Here, \(\hat{r}\) is the familiar distance operator in three dimensions introduced in this chapter. For what values of \(n\) does the corresponding Hamiltonian of such a potential have normalizable eigenstates? It's enough to just consider the case for which angular momentum is 0 .

Can you identify the \(r \rightarrow \infty\) and \(r \rightarrow 0\) behavior of a potential eigenstate?