Bound state eigenfunctions need to vanish faster than $\mathrm{\mid }x{\mathrm{\mid }}^{-3\mathrm{/}2}$

In quantum mechanics students are taught to practice that eigenfunction of a physical bound state must be continuous and vanishing asymptotically so that it is normalizable in $x\in (-\mathrm{\infty },\mathrm{\infty })$. Here we caution that such states may also give rise to infinite uncertainty in position $(\mathrm{\Delta }x=\mathrm{\infty })$, whereas $\mathrm{\Delta }p$ remains finite. Such states may be called loosely bound and spatially extended states that may be avoided by an additional condition that the eigenfunction vanishes asymptotically faster than $\mathrm{\mid }x{\mathrm{\mid }}^{-3\mathrm{/}2}$.