Uncertainty in single slit diffraction

In summary: And if the momentum uncertainty (in the y direction) at the slit would be equal to the average momentum in that direction then the diffraction pattern would have just one maximum at the center. But the diffraction pattern has a finite width, so the uncertainty in momentum must be somewhere in between. This overall uncertainty in momentum (in the y direction) is what appears as the width of the central maximum (on the screen).In summary, an electron encountering a slit of width w while moving along the x-direction with momentum p=mv will have an uncertainty in its momentum component transverse to its direction of motion, Δp. This uncertainty can be estimated by measuring the width of the central maximum of the diffraction pattern, which is related to
  • #1
Titan97
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Homework Statement


An electron is moving in a parallel beam along the x-direction with momentum, p=mv. It encounters a slit of width w. Assuming that the electron gets diffracted somewhere within the central maximum of small angular magnitude Δθ, estimate the uncertainty Δp in its momentum component transverse to the direction of motion. Check that uncertainty principle is satisfied in this experiment.

Homework Equations


ΔyΔp = ħ/2

The Attempt at a Solution


I don't know where to apply the above equation. What should be the Δy here? Is it the width w or the width Δθ?

What is the uncertainty in momentum here? If Δx is the width w, then I have to consider the Δp at the slit. (is that correct?)
 
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  • #2
Hi,
You need some more under 2. What about the width of the central maximum ? Any expressions ?
 
  • #3
the width of central maxima is obtained from ##n\lambda=w\sin\frac{\Delta\theta}{2}##

Here, n=1.
 
  • #4
##\lambda## ?
Titan97 said:
What is the uncertainty in momentum here? If Δx is the width w, then I have to consider the Δp at the slit. (is that correct?)
Yes. And you measure Δp by looking at the width of the diffraction pattern.

(you use y and x for the same direction ?)
 
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  • #5
what is delta P? the change in momentum?
 
  • #6
Yes/No. You don't have to worry about things 'before the slit'.
Assume x is horizontal and the slit is horizontal too (hyperphysics picture:)
sinslit.gif
.
His a is your w.

At the slit:

The uncertainty in position (in the y direction) is equal to the width of the slit.
And the uncertainty of the momentum (in the y direction) there (at the slit) appears as the width of the central maximum (on the screen).

[edit] picture hicks up. It's here: http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/sinslit.html##\lambda## ?
 
  • #7
BvU said:
And the uncertainty of the momentum (in the y direction) there (at the slit) appears as the width of the central maximum (on the screen).

Can you explain this?

$$\lambda=\frac{h}{mv}$$
 
  • #8
It's called the de Broglie wavelength, which google.
[edit] it is indeed the wavelength you need here. And it may seem strange, but you need the p = mv in the x-direction to determine it.
 
Last edited:
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  • #9
@BvU i am not asking about de broglie wavelength. I am asking about this

BvU said:
And the uncertainty of the momentum (in the y direction) there (at the slit) appears as the width of the central maximum (on the screen).
 
  • #10
Oh, sorry.
Well, if the momentum uncertainty (in the y direction) at the slit would be zero (the average momentum itself in that direction is zero) then there would be no diffraction pattern, just a bright line with the same height as the slit.
 
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1. What is single slit diffraction and how does it relate to uncertainty?

Single slit diffraction is a phenomenon where a wave passing through a narrow opening (or slit) spreads out, or diffracts, as it passes through the opening. This diffraction can cause uncertainty in the position of the diffracted wave, as it is spread out over a larger area than the original wave.

2. How does the size of the slit affect the uncertainty in single slit diffraction?

The size of the slit directly affects the amount of diffraction that occurs. A smaller slit will cause more diffraction, leading to a larger uncertainty in the position of the diffracted wave. Conversely, a wider slit will cause less diffraction and result in a smaller uncertainty.

3. What is the relationship between the wavelength of the wave and the uncertainty in single slit diffraction?

The wavelength of the wave also plays a role in the uncertainty of single slit diffraction. As the wavelength increases, the diffraction also increases, resulting in a larger uncertainty. This is because longer wavelengths have a harder time passing through narrow openings.

4. Can the uncertainty in single slit diffraction be calculated?

Yes, the uncertainty in single slit diffraction can be calculated using the formula Δx = λL/d, where Δx is the uncertainty, λ is the wavelength, L is the distance from the slit to the screen, and d is the width of the slit.

5. How does the uncertainty in single slit diffraction affect the overall diffraction pattern?

The uncertainty in single slit diffraction can cause the diffraction pattern to appear wider and less defined. This is because the diffracted waves are spread out over a larger area, making it more difficult to distinguish individual peaks and valleys in the pattern.

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