Bound state eigenfunctions need to vanish faster than ∣ x ∣ − 3 / 2

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