Consider any wave packet describing a free particle (so no potential or other forces acting on it). Then it can be shown that $\Delta p$ does not change in time. However, my question is what happens with $\Delta x$ as we go forward in time? Does it have to increase at all times? Or is there a counter-example where the uncertainty in position is decreasing, if even for a short time period?
My initial guess is $\Delta x$ must always increase, because $p \neq 0$, so that $\Delta p \neq0$ and hence $\Delta v = \frac{\Delta p}{m} \neq 0$. But if there's a spread in velocities, then the wave packet must also spread. Is this logic correct? Or could we have a wave packet where the back of it would move forward faster than the front, and for a certain period until that back end catches up with the front one, it would actually be narrower than at the outset, i.e. reducing $\Delta x$? If yes, how would one describe such a free particle (wave packet)?
So it does seem to me that every wave packet describing a free particle will eventually spread, but the question is whether there can be a time period in its evolution when it is actually becoming narrower.
edit: In particular, if it does not have to increase at all times, can this be shown without appealing to time reversal?
Answer
If $\Psi(x,t)$ solves the Schrodinger equation, so does $\Psi^*(x,-t)$ , so no, there is nothing at all that must increase.
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