**Unformatted text preview: **root-mean-square deviation from the mean ( i.e. standard deviation ). Note that this corresponds to a lower limit. It is always possible to know things less well. Uncertainty as a Function of Time: If at t = 0 we have localized the particle to within ∆ x , then at t = 0 ∆ p x ≈ h/ ∆ x and ∆ v x ≈ h/(m ∆ x ) . At a later time t, ∆ x = ∆ v x t ≈ ht/(m ∆ x ) and hence, 1. The better we know the particle’s position at t = 0 ( i.e. smaller ∆ x ), the worse we know it’s position at a later time t ( i.e. larger ∆ x ). 2. The uncertainty in the particle’s position ∆ x increases with time. In classical physics you can know precisely the position and momentum of a particle ( i.e. no limit on the precision). Furthermore, if you know the position and momentum of an isolated particle at t = 0, then the exact position of the particle can be predicted for all future times!...

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