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Chapter1_7 - root-mean-square deviation from the mean i.e...

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PHY4604 R. D. Field Department of Physics Chapter1_7.doc University of Florida Opposite to Classical Mechanics! Heisenberg Uncertainty Principle (1927) Due to the nature of wave superposition and wave packets we see that π 2 k x and π ω 2 t . The energy and momentum of a particle are given by x x k p E h h = = ω and hence we see that h p x x and h E t where p x is the uncertainty in the x-component of the momentum and E is an uncertainty in the energy. Thus, there is a fundamental limit on the ultimate precision with which we can know both the position (x- coordinate) of a particle and its momentum (x-component) . In addition, a measurement of a particles energy performed during a time interval t must be uncertain by an amount E . Exact Uncertainty Relations: We made a crude approximation in deriving the above relations. Later we will derive the precise form of the uncertainty relations: 2 2 2 2 h h h h E t p z p y p x z y x , where corresponds to the
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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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