Application of Heisenberg Uncertainty Principle

[Raghav Gupta]

If you challenge them, you can refer to Fowler here

Fowler is suggesting option 3 in post 25. It would be great if it is correct as I chose that option,
but @gleem Sir is saying option 1 is correct.
What option should I challenge?
It's getting confusing.

 

[BvU]

It's not called the uncertainty relation for nothing, then .

The outcomes aren't all that irreconcilable: if the exercise asks what's true for the majority of the electrons, then this carefully selected collection of electrons will indeed satisfy what you call 2 (<, formerly option C -- as you see, you yourself also contribute a little bit to the confusion ).

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.

If I were you, I wouldn't spend too much energy on this: it's simply not a very good question. Fowler (and you, and me too, and many others with us) would pick $\approx$ without hesitation. (Unless your challenging means the difference between pass or fail, in which case showing a genuine interest and a sensible defence of your choice may make a good impression and change the balance in you favour)

I would also like to point out that I strongly oppose the notion suggested in post #22:

The problem suggests that the electron in passing through the slit receive an impulse of Δp but I believe that interpretation is not warranted.

The question statement does not -- and does not have to -- suggest that at all. This whole slit business is about the wave character of the electrons: they really, intrinsically and fundamentally exhibit wave behaviour, which means that after a slit there is a diffraction pattern, even if before the slit the beam is exactly parallel (meaning infinitely wide by the same Heisenberg relation!). And the majority does end up in the central maximum.

 

[Raghav Gupta]

The outcomes aren't all that irreconcilable: if the exercise asks what's true for the majority of the electrons, then this carefully selected collection of electrons will indeed satisfy what you call 2 (<, formerly option C -- as you see, you yourself also contribute a little bit to the confusion ).

Anyways I am not challenging the problem makers, as we cannot give our evidence but only options we think correct. They might have another interpretation.
Anyways a last question,
Why this 2 or formerly C option is true for selected collection of electrons?

 

[Raghav Gupta]

Also I had a question that here in options does |p_y| is same as Δp_y?

This is an interesting video. I think question and video is same. The problematic thing were the options.

 

[BvU]

$|p_y|$ is not the same as $\Delta p_y$ but it's subtle and our language use is fuzzy (more confusion lurking!).
In principle $|p_y|$ is the expectation value of $\sqrt{p_y^2}$ and $\Delta p_y$ is the square root of the expectation value of ${p_y^2}$. Not very helpful if you aren't deep into quantum mechanics already.

Better to look at the situation described:

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).


I hope a real expert (e.g. @Orodruin) agrees somewhat -- or perhaps puts us right.
(I'm just an experimental physicist, so out on a limb -- but I like it )
---

 

[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).
---

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

--

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.