lecture6

# lecture6 - 5.61 Fall 2007 Lecture#6 page 1 WAVE-PARTICLE...

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5.61 Fall 2007 Lecture #6 page 1 WAVE-PARTICLE DUALITY of MATTER Consequences (II) ! Heisenberg Uncertainty Principle Δ x Δ p x 2 Consider diffraction through a single slit D λ = s λ D = s light, λ s l peak-null D distance x Coming out of the slit, the electrons D spread out to form a diffraction pattern with width D . x v ! v ! x v Δ This means the electrons must go through the slit with some range of velocity components v x x Now consider a beam of electrons with de Broglie wavelength λ . The slit restricts the possible positions of the electrons in the x direction: at the slit, the uncertainty in the electron x-position is xs Δ =

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! 5.61 Fall 2007 Lecture #6 page 2 Δ v D λ λ x = = v ! s Δ x Δ x Δ v x = λ v or Δ x Δ p x = λ p h But from deBroglie λ = p Δ x Δ p x = h So the position and momentum of a particle cannot both be determined with arbitrary position! Knowing one quantity with high precision means that the other must necessarily be imprecise! The conventional statement of the Heisenberg Uncertainty Principle is Δ x Δ p x 2 (depends on how “uncertainty” is defined: 1/e half-width, FWHM, etc.)
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