Application of Heisenberg Uncertainty Principle

[Raghav Gupta]

And we are into interpretation issues if we defend option 3 ($\approx$, formerly A:) which as Fowler works out, is true for the whole lot -- which is an even greater majority. The other two options (> and >>) can be ruled out.

The statement $|p_y| \;d\ < \ h \$ can be considered true for the electrons that end up in the central maximum, i.e. the majority. That central maximum has a width $\Delta p_y \approx h/d$ which is the Heisenberg uncertainty relation (see Fowler).
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Why these two statements are contradicting made by you.
In first one you are saying approx statement is for majority electrons
And in second you are saying (< ) statement is for majority?

 

[BvU]

I don't find them contradictory; why do you ?

 

[Raghav Gupta]

Means in first quote of you in post 36 you are saying,
|p_y|d ≈ h is true for majority electrons
and in second quote of you in post 36 you are saying,
|p_y|d < h is true for majority electrons?

 

[BvU]

I see what you mean: in #32 $\approx$ I should have used the $\Delta$ instead of the | | for more clarity . (I did in #36).
Case can be made that both are true (difference between $\Delta$ and | | isn't that big), so no contradiction.

 

[Raghav Gupta]

Okay got it. That Δ and | | signs were creating confusion but not now.
I wonder, why tagging @Orodruin is not catching his attention even when I see him recently on some other thread.

 

[Orodruin]

I wonder, why tagging @Orodruin is not catching his attention even when I see him recently on some other thread.

I see it. I just do not have time to read through the thread on my breaks at work.

 

[Raghav Gupta]

Twist in a tale then.
I think Hiesenberg uncertainty principle is irrelevant here after seeing this video.


Net is a wonderful thing to explore.
Got all of question and solution in this. But still why Hiesenberg gives diff. Answer and this method a different one?
About video, it is an Indian accent one but language is English. It is an approximately 1 minute video.

 

[BvU]

Total hogwash ! Re-read Fowler to understand why; he mentions it explicitly.

 

[Raghav Gupta]

Total hogwash ! Re-read Fowler to understand why; he mentions it explicitly.

But Fowler is mainly talking about uncertainty.
Here in video, the guy is saying
By De- Broglie relationship ( as Fowler also says)
λ = h/p --- 1)
Then for diffraction, λ≈d ,
But we don't want diffraction , so d >> λ
Therefore d >> h/p
So pd >> h
So what is the hogwash here?

 

[BvU]

Spelling it out (quoting Michael Fowler, Virginia univ):

we know from experiment that this is not what happens—a single slit diffraction pattern builds up, of angular width $\ \theta \sim \lambda /w$ , where the electron’s de Broglie wavelength $λ$ is given by $p_x \cong h/\lambda$ (there is a negligible contribution to $λ$ from the y-momentum). The consequent uncertainty in $p_y$ is

$$Δp_y/p_x \cong \theta \cong \lambda/w$$

Putting in $p_x = h/\lambda$ , we find immediately that

$$\Delta p_y = h/λ$$

I can't put it into words any better than that

--

 

[Raghav Gupta]

Spelling it out (quoting Michael Fowler, Virginia univ):

I can't put it into words any better than that

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That is correct,
Δp_y=h/λ
Then how the term d will be introduced?

 

[Raghav Gupta]

That is what I have written in post 44,
λ = h/p
Manipulating,
p = h/λ

 

[BvU]

Connect the dots: Fowler's w is your d.

 

[Raghav Gupta]

Connect the dots: Fowler's w is your d.

Yeah, got it from that
Δp_yd ≈ h , thanks.
But I should admit
2 mistakes
First from the answer key of our paper
And then from the video solution.

 

[BvU]

I'm pretty convinced the answer $\ |p_y|\;d \ < \ h\$ is actually correct

 

[Raghav Gupta]

I'm pretty convinced the answer $\ |p_y|\;d \ < \ h\$ is actually correct

Haha,
You were saying the other thing previously.

 

[BvU]

Are we back to posts 32, 35, 36 and have to go the loop again ?

 

[Raghav Gupta]

Are we back to posts 32, 35, 36 and have to go the loop again ?

Yeah, I think
There is confusion between two options
|p_y|d < h
|p_y |d ≅ h

 

[Raghav Gupta]

Are we back to posts 32, 35, 36 and have to go the loop again ?

Sorry for the post 53, it was non sensible.
Can you tell, if we have got from fowler Δp_yd ≈h
Then how | p_y| d < h ?
I have seen posts 32,35,36 carefully now.
You are saying this is not good exercise but anyways I am interested.

 

[Raghav Gupta]

(difference between $\Delta$ and | | isn't that big)

This statement was also a bit confusing from you for me
As I replaced Δp_yd ≈h
To |p_y|d ≈h
Because you are saying difference between these two signs are not big.

 

[BvU]

Sorry for the post 53, it was non sensible.
Can you tell, if we have got from fowler Δp_yd ≈h
Then how | p_y| d < h ?
I have seen posts 32,35,36 carefully now.
You are saying this is not good exercise but anyways I am interested.

The central maximum in the diffraction pattern from a single slit has a width Δp_y ≈ h / d (see Fowler). Most electrons end up in the central maximum, so for most electrons | p_y| d < h

I am having a bit of a deja vu feeling now.

 

[Raghav Gupta]

Okay, got it but not completely.
I have not read diffraction so much. I guess I have to know some very basic concepts first to understand whole of the fowler.
Will search myself for the moment.
Thanks by the way.

 

[BvU]

You're welcome. Don't forget #32: don't spend too much energy on this. It'll come by a few more times later on in the curriculum in different incarnations.