Using diffraction slit as a lens

[gildomar]

Homework Statement

"A TV tube manufacturer is attempting to improve the picture resolution, while keeping
costs down, by designing an electron gun that produces an electron beam which will make
the smallest possible spot on the face of the tube, using only an electron emitting cathode
followed by a system of two well-spaced apertures. (a) Show that there is an optimum
diameter for the second aperture. (b) Using reasonable TV , tube parameters, estimate the
minimum possible spot size."

Homework Equations

ΔpΔx≥$\frac{\hbar}{2}$

a sin θ = mλ ?

The Attempt at a Solution

I was thinking that I need to use single slit diffraction on this, since while the single slit equation is for the minimum on the screen, if it goes an equal distance past that, it will create a maxima. I'm a little fuzzy on how Heisenberg's equation fits, since this was from the chapter on the uncertainty principle.

p.s. I was unsure if this question should go here, or in the introductory questions section. I thought that quantum mechanics should be directed here though.

 

[mark726]

Yes, you are on the right track. The Heisenberg uncertainty principle states that the uncertainty in the position of an electron (Δx) multiplied by the uncertainty in its momentum (Δp) must be greater than or equal to Planck's constant divided by 2 (h/2). This means that the smaller the spot size of the electron beam, the greater the uncertainty in its momentum. So, to minimize the spot size, we need to minimize the uncertainty in its momentum.

In this case, the momentum of the electron is related to its wavelength by the de Broglie relation: p = h/λ. So, to minimize the uncertainty in its momentum, we need to minimize the uncertainty in its wavelength. This is where the single slit diffraction comes in. As you mentioned, the single slit diffraction equation can also be used to determine the minimum spot size of the electron beam. The equation is: Δx = λL/d, where L is the distance from the aperture to the screen and d is the width of the aperture.

To find the optimum diameter for the second aperture, you need to find the value of d that will minimize the spot size. This can be done by setting the derivative of the single slit diffraction equation equal to 0 and solving for d. This will give you the optimum diameter for the second aperture.

To estimate the minimum possible spot size, you can use reasonable TV tube parameters, such as the distance from the aperture to the screen, the electron energy, and the diameter of the first aperture. You can then plug these values into the single slit diffraction equation to calculate the minimum spot size.

I hope this helps! Let me know if you have any further questions.