Double-slit Experiment: Use the uncertainty principle to show that...

[mario99]

In a double-slit experiment on electrons (or photons), suppose that we use indicators to determine which slit each electron went through. These indicators must tell us the y coordinate to within $d/2$, where $d$ is the distance between slits. Use the uncertainty principle to show that the interference pattern will be destroyed [Note: First show that the angle $\theta$ between maxima and minima of the interference pattern is given by $\frac{1}{2}\lambda/d.$]



Any help would be appreciated.

 

[topsquark]

In a double-slit experiment on electrons (or photons), suppose that we use indicators to determine which slit each electron went through. These indicators must tell us the y coordinate to within $d/2$, where $d$ is the distance between slits. Use the uncertainty principle to show that the interference pattern will be destroyed [Note: First show that the angle $\theta$ between maxima and minima of the interference pattern is given by $\frac{1}{2}\lambda/d.$]



Any help would be appreciated.

This is Physics, not Calculus.

So far as I know, the Uncertainty Principle has nothing to do with this. Where did you get this problem from?

-Dan

 

[khansaheb]

In a double-slit experiment on electrons (or photons), suppose that we use indicators to determine which slit each electron went through. These indicators must tell us the y coordinate to within $d/2$, where $d$ is the distance between slits. Use the uncertainty principle to show that the interference pattern will be destroyed [Note: First show that the angle $\theta$ between maxima and minima of the interference pattern is given by $\frac{1}{2}\lambda/d.$]



Any help would be appreciated.

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

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Please share your work/thoughts about this problem.

 

[mario99]

Thank you topsquark and khansaheb for helping me.

This is Physics, not Calculus.

So far as I know, the Uncertainty Principle has nothing to do with this. Where did you get this problem from?

-Dan

I'm sorry to say this, but you have a misconception between Physics and Calculus. It is wrong to say this is physics, not calculus. Calculus is just a tool to solve physics' problems.

The maxima and minima suggests that there is calculus in this physics problem.

I have some questions for you:

1- What is the relationship between uncertainty principle and interference pattern?
2- Why a destruction will occur when the angle between maximum and minimum patter is $\frac{1}{2}\lambda/d$?

Answer theses questions and we will have more talks.

 

[mario99]

Where did you get this problem from?

-Dan

I got this problem from the bakery when I was buying some bread. I am just kidding. I got it from my teach who probably got it from the book. You can find this problem, or at least a similar one, in any physics book in the Quantum Mechanics chapter.

 

[The Highlander]

I got this problem from the bakery when I was buying some bread. I am just kidding. I got it from my teach who probably got it from the book. You can find this problem, or at least a similar one, in any physics book in the Quantum Mechanics chapter.

So, it is Physics then.

 

[mario99]

So, it is Physics then.

?

 

[topsquark]

Thank you topsquark and khansaheb for helping me.




I'm sorry to say this, but you have a misconception between Physics and Calculus. It is wrong to say this is physics, not calculus. Calculus is just a tool to solve physics' problems.

The maxima and minima suggests that there is calculus in this physics problem.

I have some questions for you:

1- What is the relationship between uncertainty principle and interference pattern?
2- Why a destruction will occur when the angle between maximum and minimum patter is $\frac{1}{2}\lambda/d$?

Answer theses questions and we will have more talks.

I'm sorry, but not everything in Physics involves Calculus; not everything that involves a max and min has anything to do with Calculus, either. Honestly, if you had read the relevant section in your textbook on this you would realize that there are no integrals, nor derivatives, nor infinitesimals at all in this problem.

And, again, the UP has nothing to do with the interference pattern. The interference pattern is strictly due to the observation/non-observation of the particles as they pass through the slits.

Give me the reference to this problem from any Physics text that you like: I would be fascinated to read it. I've never seen anyone mention a connection between the UP and the double slit experiment before. (And, just for the record, I own 9 texts on QM that discuss the double slit experiment.)

To address your second question, the wave from one slit has to travel a slight distance longer than the wave from the other slit. This means that there is a phase difference between the two waves when they meet at the screen. Can you tell me what the phase difference has to be between two superimposed waves to create total destructive interference?

-Dan

 

[Dr.Peterson]

I'm sorry, but not everything in Physics involves Calculus; not everything that involves a max and min has anything to do with Calculus, either. Honestly, if you had read the relevant section in your textbook on this you would realize that there are no integrals, nor derivatives, nor infinitesimals at all in this problem.

And, again, the UP has nothing to do with the interference pattern. The interference pattern is strictly due to the observation/non-observation of the particles as they pass through the slits.

Give me the reference to this problem from any Physics text that you like: I would be fascinated to read it. I've never seen anyone mention a connection between the UP and the double slit experiment before. (And, just for the record, I own 9 texts on QM that discuss the double slit experiment.)

To address your second question, the wave from one slit has to travel a slight distance longer than the wave from the other slit. This means that there is a phase difference between the two waves when they meet at the screen. Can you tell me what the phase difference has to be between two superimposed waves to create total destructive interference?

-Dan

For what it's worth, I find this problem on several sites, and it does appear to come from a textbook. (I haven't read through any.)

Note that it doesn't say the UP causes the interference pattern, but that somehow it would destroy the pattern.

 

[topsquark]

For what it's worth, I find this problem on several sites, and it does appear to come from a textbook. (I haven't read through any.)

Note that it doesn't say the UP causes the interference pattern, but that somehow it would destroy the pattern.

Okay, reading through some of this I think I know what the problem might be getting at, though I'll say up front that I don't agree with it. I'll explain that at the end.

I think the argument is supposed to go like this:

Say we have a light source that is giving off light waves for at least a short period of time. Over that period, if we do not try to detect the particle (at either slit) then the waveform spreads out, creating an interference pattern on the screen. If we detect which slit the particle goes through (let's say for the sake of argument we have a detector on each slit) then we know where the particle should be at some point, which means that the momentum will be very uncertain. Now, because the particle is localized, it's wavefunction will be peaked and there is nothing between the slits and screen to change the momentum part of the wavefunction, so the wave stays peaked. Thus, there is no interference pattern on the screen.

BUT
1) We don't really need the UP for this. All we need is the fact that the wave peaks at the detector. Time evolution will "spread out" the wavefunction, but over a short distance this will not really have an effect on the pattern. We still destroy the interference pattern.

2) Nothing of what said above explains why, if we take away the detectors and only shoot one particle at a time at the slits, why we build up an interference pattern on the screen if we do that a large number of times, and why that pattern doesn't show up if we detect which slit the particle goes through.

IMHO the best way to explain the double slit experiment is the axiom that a particle cannot be said to be in any particular state when we are not measuring it and (barring time evolution) will stay in that state until something interacts with it. The wavefunction will thus initially spread out through all of space creating an interference pattern, and if it goes through a detector the wavefunction "collapses" into a state with a peaked position wavefunction, then proceeds to the screen and does not create interference. The UP just confuses the issue and doesn't really add anything of substance.

-Dan