Uncertainty in single slit diffraction

[Titan97]

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?)

 

[BvU]

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

 

[Titan97]

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

Here, n=1.

 

[BvU]

$\lambda$ ?

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 ?)

 

[Titan97]

what is delta P? the change in momentum?

 

[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:)

.
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$\lambda$ ?

 

[Titan97]

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?

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

 

[BvU]

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.

 

[Titan97]

@BvU i am not asking about de broglie wavelength. I am asking about this

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).

 

[BvU]

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.