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

Abstract

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 $x\in (-\infty, \infty)$. Here we caution that such states may also give rise to infinite uncertainty in position $(\Delta x=\infty)$, whereas $\Delta p$ 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 $|x|^{-3/2}$.