Uncertanty in double slit diffraction

[darthmonkey]

In a double-slit experiment electrons are sent through a doule slit where an indicator determines the slit each electron went through. These indicators tell the y coordinate to within d/2, where d is the distance between the slits. If this is the case show that the diffraction pattern will be destroyed.


The angular distance,Δθ, between a maximum and minimum is λ/(2d). The angular distance may also be approximated as Δy/l where l is the distance from the slits to the screen. The y seperation,Δy, at the screen between a max and adjacent minimum is (λl)/(2d).

I understand that in order for the pattern to be destroyed the uncertainty in the y position at the screen must be equal to the distance between adjacent min and max, but I can't seem to get there.

The uncertainty of y at the slits is given in the problem as d/2 allowing the momentum in y to be found. Assuming the uncertainty in p at the slit is the same at the screen then Δp is known. Assuming the electron has a momentum in the x direction then the momentum in x and the wavelength may be used. I tried to say that Δθ=Δy/l, and then substitute the p relation in for y but this gets me no where. Messing around I found out that setting Δy/l=Δp_y/p_x and solving for Δy gives the correct relation but I have no idea why you can set these two things equal to each other.

 

[jbsteele]

To answer this question, we first need to calculate the uncertainty in the y coordinate at the screen. The uncertainty in the y coordinate at the screen is given by the Heisenberg Uncertainty Principle as: Δy x Δp = h/2π. Since we know the uncertainty in the y coordinate at the slits (d/2), we can use this to calculate the uncertainty in the momentum at the slits: Δp = h/(2πd). We can now use this to calculate the uncertainty in the y coordinate at the screen: Δy = (h/(2πd)) x l/p, where l is the distance from the slits to the screen and p is the momentum of the electrons. Now, we can compare this uncertainty in the y coordinate to the distance between adjacent maxima and minima. As we mentioned earlier, the angular distance between a maximum and minimum is λ/(2d). This is also equal to Δy/l, where Δy is the distance between a maximum and adjacent minimum at the screen. Therefore, if the uncertainty in the y coordinate at the screen is greater than or equal to Δy/l, then the diffraction pattern will be destroyed. In other words, if Δy > (h/(2πd)) x l/p, then the diffraction pattern will be destroyed.