Uncertainty Principle beam of electrons

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Homework Statement


A beam of 50eV electrons travel towards a slit of width 6 micro metres. The diffraction pattern is observed on a screen 2m away.

Use the angular spread of the central diffraction pattern (+/- lambda / slit width) to estimate the uncertainty in the y-component of momentum of an electron.

Use this result and the Heisenberg uncertainty principle to estimate the minimum uncertainty in the y-coordinate of an electron just after it has passed trough the slit. Comment on this result.

Homework Equations


n[tex]\lambda[/tex] = d sin[tex]\theta[/tex] [1]
[tex]\lambda[/tex] = [tex]\frac{h}{p}[/tex] [2]
E = [tex]\frac{p^{2}}{2m}[/tex] [3]

The Attempt at a Solution



Okay, well I first began by using equation 3 combined with equation 2 to work out the wavelength of the electron. This came out as 1.74 * 10^-10 m.

Re-arranging equation 1 i get [tex]sin\theta[/tex] = +/- lambda/d = +/- 2.9*10^-5

Now i get to the crux of my problem

QM.jpg


It asks to work out the uncertainty in the y component of the momentum, is the uncertainty in the y position given by what's labelled there in the picture, or is it from the central point to the top of the fringe?

Assuming that it is what i've drawn, that leads me to the uncertainty in momentum being = 4.55*10^-31. Well okay.

However again i'm troubled by the next part, it says using the result of the past bit...I can't see how they relate at all! :(. Also i can't see how the further away the screen gets, the more certain the momentum will become! I'm confused.

Thanks for reading this far !

I appreciate any help you can offer!
 

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