# Interpretation of Dispersion Behavior of Wave Function

Hi,

I've been reading a QM book and it mentions that particles can be represented as a wave packet, which provides a description for particles simultaneously as a wave and particle.

It also mentions that the wave packets disperse, and the width becomes extremely large for free microscopic objects in a short period of time.

What is the physical interpretation of this?
Does this mean if I create a free electron at rest, it would be impossible for me to detect it's position after, say 5 minutes, since the probability density everywhere is now so small after the dispersion?

Thanks.

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Nugatory
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What is the physical interpretation of this?
Does this mean if I create a free electron at rest, it would be impossible for me to detect it's position after, say 5 minutes, since the probability density everywhere is now so small after the dispersion?
Not quite. After five minutes, you can still detect it and even state the exact location where you detected it. What's impossible is to predict what that location will be without actually doing the measurement to detect the electron.

Not quite. After five minutes, you can still detect it and even state the exact location where you detected it. What's impossible is to predict what that location will be without actually doing the measurement to detect the electron.
I'm a bit confused.

For example, suppose the initial width of the wave packet was 1mm, then I have a very good chance of finding the particle within 1mm of the original position I created it since the probability is very high in that region.

After it dispersed, shouldn't the probability be so low in that 1mm region such that I would almost have no chance of finding it? Of course I could still "eventually" find it somewhere and then state it's position, but shouldn't the probability of finding it in a random measurement with an uncertainty of 1mm be extremely low?

Thanks again.

Nugatory
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After it dispersed, shouldn't the probability be so low in that 1mm region such that I would almost have no chance of finding it? Of course I could still "eventually" find it somewhere and then state it's position, but shouldn't the probability of finding it in a random measurement with an uncertainty of 1mm be extremely low?
Yes, that's right. But we can imagine (this being a thought experiment and all) ways of performing the measurement that don't have this problem. For example, we could take a screen with a 1mm mesh, mount a tiny electron detector in each opening in the mesh, and sweep it through a very large volume of space, see which detector triggers.

The important point here is that we shouldn't be thinking of in terms of "the electron is somewhere, we just don't know where yet" the way that I think about the small and expensive machine part that I just dropped and watched bounce off into some remote corner of my shop. The electron, both in the math and in many of the interpretations of QM, isn't anywhere until one of the detectors triggers... And the wave function isn't telling us the probability of the electron being at a particular point, it's telling us the probability that the detector at that point will trigger.

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