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

TL;DR

This paper discusses the asymptotic decay rate of bound state eigenfunctions in quantum mechanics, proposing that they must vanish faster than |x|^{-3/2} to ensure finite position uncertainty.

Contribution

It introduces a stricter condition on eigenfunction decay rates, emphasizing the importance of faster-than-|x|^{-3/2} decay for physically meaningful bound states.

Findings

Eigenfunctions with slower decay can lead to infinite position uncertainty.

Bound states should decay faster than |x|^{-3/2} to be physically acceptable.

Loosely bound states may have finite momentum uncertainty but infinite position uncertainty.

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