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This question already has an answer here:

For the example of an infinite square well, $\psi(x)=0$ for $x$ outside the well/interval, and we are to interpret this as the particle cannot be found outside the well because $|\Psi(x,t)\bar{\Psi}(x,t)|^2=0$ in these regions. But probability of $0$ does not necessarily imply that an event is impossible, so I'm a bit confused as to why you can say that it's impossible for the particle to be in this region or that just based off of this. Can anyone clarify?

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marked as duplicate by ACuriousMind, Danu, Brandon Enright, Kyle Kanos, JamalS Nov 13 '14 at 7:47

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When dealing with the infinite square well, we must be clear that it is a limit of the finite square well case. But even though for the finite case we have as Hilbert space $L^2(\Bbb{R})$, that is, the particle can have non zero probability of being found in any region of non-zero measure, for the infinite case, the limit forces the condition of working with the Hilbert space $L^2[0,a]$. In this case the domain of the wavefunction $\psi(x)$ is $[0,a]$, so, it is indeed imbossible to find the particle outside the well because these events are not acceptable, since any region outside $[0,a]$ is not an element of the space of possible events (or in a more precise language of the $\sigma$-algebra) and not because they have zero probability.

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  • $\begingroup$ Ahh, thank you. This is the type of answer I was looking for :) $\endgroup$ – user153582 Nov 14 '14 at 5:52
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Why "probability zero doesn't imply that an event is impossible"? Low probability implies that an event is possible, but improbable. But probability ZERO means that the event is IMPOSSIBLE, not IMPROBABLE.

The wave-function doesn't cheat. There where the wave-function is zero, the presence of the particle is FORBIDDEN.

Good luck !

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  • $\begingroup$ See, that's my confusion though. Probability zero doesn't mean that an event is impossible. If you throw a dart at a square, and the only possibility is that the dart lands on a single point, the probability of the dart landing on any one point is zero (a point has zero area), yet, you throw the dart and it lands on a point. So something of probability zero just happened. $\endgroup$ – user153582 Nov 14 '14 at 5:49
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It seems to me your question is on the equivalence of possibility with probability, and not the infinite quantum well. Maybe I am being sloppy, but I would use them interchangeably in the following context: If probability of something is absolute zero, it means no matter how many times you try the experiment (looking for a particle outside the well), you will not find it. Put it mathematically: $$N_{outside}=Prob(outside)\times N_{tries} \\ =0\times N_{tries}=0 $$ no matter how large the $N_{tries}$. Therefore, statistically speaking, it seems safe to say it is impossible to find the particle outside the well.

Besides, if the well is really infinite, the particle would need infinite energy to overcome it. Now that is really impossible.

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This exercise is a mathematical construct and infinite potential really means the probability of finding the particle outside is zero. It will never ever be outside the well in this scenario.

In a finite case, the probability of finding the particle outside of the well goes down exponentially, thus its never really zero, only very small. This is what you mean when you say its not impossible, just unlikely.

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