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Uncertainty and double slits

  • Thread starter kof9595995
  • Start date
679
2
1. Homework Statement
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. Homework 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. Homework Statement



2. Homework Equations



3. The Attempt at a Solution
 
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