Consequences of the Uncertainty Principle The deBroglie relation λ = h/p and the uncertainty principle Δ p x Δ x ≥ ¯ h/ 2, which reﬂect the wavelike nature of matter, imply that a localized particle (in an atom or a nucleus or passing through a slit, etc.) cannot have zero kinetic energy. Three ways to see this (there are more!): 1. From the uncertainty principle, if a particle is conﬁned to Δ x , the momentum will be at least Δ p x = ¯ h/ (2Δ x ), where ¯ h = h/ 2 π . 2. If a particle with initial momentum p x = p and p y = 0 passes through a slit of width d , it will di±ract, which means it spreads out in the y direction. So localizing in the y direction makes p y > 0. Estimate: The angle to the ﬁrst minimum is given by λ = d sin θ and sin θ ≈ p y /p . The deBroglie relation λ = h/p gives the uncertainty principle result with δy ≈ d and δp y ≈ p y . 3. The wavelength of a particle cannot be much larger than the size of the region of localization. From p = h/λ
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