Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 4 / 15D / b / i / Solution

For a bound state, $E<0$ and therefore $\gamma >0$. As $r\to \infty$, the terms proportional to $1/r$ are subleading and the radial Schrodinger equation has dominant balanceIts exponential behaviours are $e^{\gamma r}$ and $e^{-\gamma r}$. The growing solution is not normalizable, so

Solved by gpt-5.6-sol high.