Uncertainty Principle beam of electrons

[EEWannabe]

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$$\lambda$$ = d sin$$\theta$$ [1]
$$\lambda$$ = $$\frac{h}{p}$$ [2]
E = $$\frac{p^{2}}{2m}$$ [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 $$sin\theta$$ = +/- lambda/d = +/- 2.9*10^-5

Now i get to the crux of my problem

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!

 

[anathema]

Thank you for your post. I understand your confusion and I would be happy to help clarify the concepts for you.

Firstly, let's review the equations you have used. Equation 1 is the diffraction equation, which relates the wavelength of a wave (in this case, the electron) to the distance between the slits and the angle at which the diffraction pattern is observed. Equation 2 is the de Broglie wavelength equation, which relates the wavelength of a particle to its momentum. And equation 3 is the kinetic energy equation, which relates the kinetic energy of a particle to its momentum and mass.

Now, to address your first question about the uncertainty in the y component of momentum. The uncertainty in momentum is not related to the diffraction pattern itself, but rather to the Heisenberg uncertainty principle. This principle states that it is impossible to simultaneously know the exact position and momentum of a particle. As you correctly pointed out, the uncertainty in the y component of momentum can be calculated using the formula you have derived from equation 1. However, this only gives you the uncertainty in momentum due to the diffraction pattern.

To calculate the minimum uncertainty in the y-coordinate of an electron just after it has passed through the slit, we need to use the Heisenberg uncertainty principle. This principle tells us that the product of the uncertainties in position and momentum must always be greater than or equal to a certain value, known as Planck's constant (h). So, in this case, the minimum uncertainty in the y-coordinate can be calculated as h/2∆py, where ∆py is the uncertainty in the y component of momentum.

Now, to address your second question about the relationship between the screen distance and the certainty of momentum. As the screen distance increases, the diffraction pattern becomes narrower and the fringes become sharper. This means that the uncertainty in the y component of momentum decreases, leading to a more certain measurement. This is because as the distance increases, the angle at which the diffraction pattern is observed becomes smaller, and thus the uncertainty in momentum (calculated using equation 1) decreases.

I hope this helps to clarify the concepts for you. Please let me know if you have any further questions.