 Solution

For brevity write $\gamma =mKZ/{\hslash }^2$. With the physical indicial root $c=\ell +1$, substitute $f={\sum }_{n\ge 0}{a}_n{r}^{n+\ell +1}$ into the equation from part (i). The coefficient of $r^{n+\ell }$ isThe indicial equation has already canceled the still lower power $r^{\ell -1}$. Rearranging gives the recurrence relationIts denominator is $(n+1)(n+2\ell +2)>0$ for every $n\ge 0$.

To impose wavefunction normalization at infinity, first suppose that the series never terminates. ThenThis is the exponential-series growth associated with $e^{2\alpha r}$, which would overwhelm the factor $e^{-\alpha r}$ in $\psi$. A direct lower-bound argument makes the conclusion precise. All sufficiently late recurrence ratios are real and positive, so after a common global phase choice the tail coefficients have one sign. For any $0<\eta <\alpha$, there is an index $N$ such thatInduction then bounds the tail below by a positive multiple ofThe finitely many earlier terms form a polynomial and cannot cancel this exponential growth. Thus $|\psi |=r^{-1}{e}^{-\alpha r}|f|$ grows at least as a positive multiple of $r^{\ell }{e}^{(\alpha -\eta )r}$ for large $r$, contradicting wavefunction normalization. This proves that the regular series must terminate.

Since the denominator never vanishes, termination requires the numerator to vanish at some nonnegative integer $n$:Then $a_{n+1}=0$ and every later coefficient is zero. Consequently $f$ is a polynomial of degree $n+\ell +1$, with a factor $r^{\ell +1}$; equivalently,where $P_n$ has degree $n$. This is series-termination quantization of the Coulomb radial equation. If $N_{\mathrm{p}}=n+\ell +1$ denotes the principal quantum number, the associated energy eigenvalues areHere $n$ counts the degree of the remaining radial polynomial; it is not the principal quantum number itself.