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Morrison writes in "Morrison, Michael A. : Understanding quantum physics : a user's manual"

$ |\Psi(x,t)|^2 \xrightarrow[x\rightarrow\pm \infty ]{} 0$ at all times t [bound state]

$ |\Psi(x,t)|^2 \xrightarrow[x\rightarrow\pm \infty ]{} 0$ at any particular time t [unbound state]

So I can imagine that "all" means the entirety of all times, but do I not get "all" when summing over all particular states?

I also understand that in a bound state, the wave is never at the infinity position, but the wave of an unbound state may exist there.

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1 Answer 1

I don't have access to the book M.A. Morrison, Understanding quantum physics: a user's manual, but it seems that the phrase

at all times $t$

is used in the sense that the $x$-limit is

uniform in $t\in\mathbb{R}$,

while the phrase

at any particular time $t$

is used in the sense that the $x$-limit is

pointwise in $t\in\mathbb{R}$.

In other words, the difference is in the ordering of pertinent quantifiers. Note that Morrison's above terminology is not standard.

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This is surely right, but one could clarify by saying that a nonuniform limit happens when a bump is travelling to infinity without spreading out. This doesn't happen in the Schrodinger equation, so he could have been less careful. –  Ron Maimon Aug 22 '12 at 6:31

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