Probability at Minima or Maxima

[VenomHowell15]

I hate to make another topic so soon, but I'd like to get some opinions on this question...

Homework Statement

A two slit electron difraction experiment is done with slits of unequal widths. When only slit 1 is open, the number of electrons reaching the screen per second is 25 times the number of electrons reaching the screen per second when only slit 2 is open. When both slits are open, an interference pattern results in which the destructive interference is not complete. Find the ratio of the probability of an electron arriving at the interference maximum to the probability of an electron arriving at an adjacent interference minimum.

Homework Equations

Superposition principle, of course.

|(psi1)^2 +(psi2)^2| (?)

Maxima at (theta) = h/(pD) (?)
Minima at (theta) = h/(2pD) (?)

Group velocity and/or Phase Velocity?

The Attempt at a Solution

According to the book, the solution should be 2.25:1. Problem is, I don't quite know where to begin with the question. I'm running on practically no sleep right now and probably am just skipping over something to kick off the question, so if I could just get a little nudge in the right direction I could probably figure this out on my own.

 

[Jhero]

Thank you for your question. It seems like you are struggling with understanding the concept of interference in a two-slit electron diffraction experiment. Let me try to explain it to you in simpler terms.

In this experiment, electrons are being sent through two slits and then hitting a screen behind the slits. When only slit 1 is open, the number of electrons hitting the screen per second is 25 times the number when only slit 2 is open. This means that slit 1 is much wider than slit 2, allowing more electrons to pass through.

Now, when both slits are open, an interference pattern is formed on the screen. This happens because the electrons that pass through each slit interfere with each other. This interference can be constructive (where the waves add up) or destructive (where the waves cancel each other out).

In this case, the interference pattern is not completely destructive, which means that some electrons are still reaching the screen. This can be explained by the superposition principle, which states that the total wave function is the sum of the individual wave functions from each slit.

To find the ratio of the probability of an electron arriving at the interference maximum to the probability of an electron arriving at an adjacent interference minimum, we need to look at the equations for the maxima and minima in the interference pattern.

The maxima occur at (theta) = h/(pD), where h is the Planck's constant, p is the momentum of the electron, and D is the distance between the slits and the screen. The minima occur at (theta) = h/(2pD).

Now, let's consider the probability of an electron arriving at the interference maximum. This can be calculated by taking the square of the total wave function at that point. So, the probability is proportional to |(psi1)^2 +(psi2)^2|.

Similarly, for an electron arriving at an adjacent interference minimum, the probability is proportional to |(psi1)^2 -(psi2)^2|.

To find the ratio, we can divide these two probabilities, which gives us |(psi1)^2 +(psi2)^2| / |(psi1)^2 -(psi2)^2|. Using the given information, we can substitute the values of (psi1)^2 and (psi2)^2 to get the ratio of 2.25:1.

I hope this helps and clears up any confusion.