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Uncertainty and double slits

  1. Sep 14, 2009 #1
    1. The problem statement, all variables and given/known data
    Show that in order to be able to determine through which slit a double slit system each photon passes without destroying the double silt diffraction pattern, the condition
    [tex]\delta y\delta {p_y} \ll \frac{h}{{4\pi }}[/tex]
    must be satisfied. Since this condition violated the uncertainty principle, it can not be met

    2. Relevant equations
    [tex]d\sin \theta = \lambda [/tex]
    [tex]\sin \theta = \frac{{\delta {p_y}}}{p}[/tex]
    [tex]\lambda = \frac{h}{p}[/tex]



    3. The attempt at a solution
    In order not to destroy the pattern, the angle should not be large enough to shift one maximum to its adjacent maximum. So [tex]\sin \theta \ll \frac{\lambda }{d}[/tex], and then we have
    [tex]\frac{{\delta {p_y}}}{p} \ll \frac{\lambda }{d}[/tex]
    To figure out which slit the photon passes through, we must have
    [tex]\delta y \ll \frac{d}{2}[/tex]
    Combine these two and use de broglie's relation [tex]\lambda = \frac{h}{p}[/tex]
    We can get
    [tex]\delta y\delta {p_y} \ll \frac{h}{2}[/tex]

    But it seems to me the extra [tex]2\pi [/tex] just comes out from nowhere. I'm really pulling my hair off on this quetsion
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Sep 15, 2009
  2. jcsd
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