Explain qualitatively that the uncertainty in position of a partcile increases t

Question

Explain qualitatively that the uncertainty in position of a partcile increases the more accurately we localize the particle initially and that the uncertainty increases with time.

Explanation / Answer

The uncertainty principle tells us that: Dx Dt >= hbar/2 . The equality occurs only for a Gaussian shaped spatial probability distribution. You can check that the momentum probability distribution is also gaussian in this case. You can find out the relation between the spatial width and the width of the momentum spectrum. Since all wave components don't have the same momentum, the wave packet will spread, of course. You could try to calculate that, this is easy since it is the solution of the SE in free space.

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