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 $|x{|}^{-3/2}$.

One Fig. (3 parts), 6 pages, Text around Eq. (13,14) modified

References in corpus (2)

• Experimental demonstration of a universally valid error-disturbance uncertainty relation in spin-measurements
• Position-momentum uncertainty products

Cited by in corpus (1)

• Divergence of $⟨p^6⟩$ in discontinuous potential wells