# Uncertainty in single slit diffraction

Tags:
1. Sep 9, 2016

### Titan97

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
ΔyΔp = ħ/2

3. 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?)

2. Sep 9, 2016

### BvU

Hi,
You need some more under 2. What about the width of the central maximum ? Any expressions ?

3. Sep 9, 2016

### Titan97

the width of central maxima is obtained from $n\lambda=w\sin\frac{\Delta\theta}{2}$

Here, n=1.

4. Sep 9, 2016

### BvU

$\lambda$ ?
Yes. And you measure Δp by looking at the width of the diffraction pattern.

(you use y and x for the same direction ?)

Last edited: Sep 9, 2016
5. Sep 9, 2016

### Titan97

what is delta P? the change in momentum?

6. Sep 9, 2016

### BvU

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:)
.

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).

 picture hicks up. It's here: http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/sinslit.html

$\lambda$ ?

7. Sep 9, 2016

### Titan97

Can you explain this?

$$\lambda=\frac{h}{mv}$$

8. Sep 9, 2016

### BvU

It's called the de Broglie wavelength, which google.
 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: Sep 9, 2016
9. Sep 9, 2016