束缚态本征函数需要比$|x|^{-3/2}$更快地消失
Bound state eigenfunctions need to vanish faster than $|x|^{-3/2}$
摘要：
在量子力学中，学生被教导练习物理束缚态的本征函数必须是连续的，并且渐近地消失，这样它才能在$x\in (-\infty, \infty)$中归一化。这里我们要注意，这样的状态也可能引起位置$(Δx=\infty)$的无限不确定性，而$Δp$仍然是有限的。这样的状态可以被称为松散束缚和空间扩展状态，其可以通过本征函数比$|x|^{-3/2}$更快地渐近消失的附加条件来避免。
英文摘要：
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 $(Δx=\infty)$, whereas $Δp$ 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 $|x|^{-3/2}$.
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