Is the Bohr Model Defined by Equating Coulomb and Centripetal Forces?

[drop_out_kid]

Homework Statement

(a) Determine the frequency of the orbital motion of an electron in a Bohr-model hydrogen
atom for a level with the quantum number n. (b) Calculate the frequency of the radiation
emitted in the transition from the state n to the state n-1. (c) Show that the results of (a)
and (b) agree if n is very large. (This is an important example of Bohr’s correspondence
principle.)

Relevant Equations

Rydberg formula, mvr=nh_bar

So for this question I just want to make sure that
1. Bohr model is that F_coulomb = F_centripetal? and then get w(r) is called determind?
2. for (b) calculate the frequency, should I use Rydberg formula or what?

 

[kuruman]

Homework Statement:: (a) Determine the frequency of the orbital motion of an electron in a Bohr-model hydrogen
atom for a level with the quantum number n. (b) Calculate the frequency of the radiation
emitted in the transition from the state n to the state n-1. (c) Show that the results of (a)
and (b) agree if n is very large. (This is an important example of Bohr’s correspondence
principle.)
Relevant Equations:: Rydberg formula, mvr=nh_bar

So for this question I just want to make sure that
1. Bohr model is that F_coulomb = F_centripetal? and then get w(r) is called determind?
2. for (b) calculate the frequency, should I use Rydberg formula or what?

If you want to receive help, I think you going to have to show some work here, like calculating the frequency of orbital motion required for part (a). Yes, a good place to start is F_coulomb = F_centripetal. I would worry about part (b) later.

 

[drop_out_kid]

If you want to receive help, I think you going to have to show some work here, like calculating the frequency of orbital motion required for part (a). Yes, a good place to start is F_coulomb = F_centripetal. I would worry about part (b) later.

Do you think is this what problem asks and is it right? also it seems have missed a 1/2 I am not sure:

 

[drop_out_kid]

If you want to receive help, I think you going to have to show some work here, like calculating the frequency of orbital motion required for part (a). Yes, a good place to start is F_coulomb = F_centripetal. I would worry about part (b) later.

How can I add a pic tho. I found the picture gone I will add in the post reply

 

[kuruman]

Do you think is this what problem asks and is it right? also it seems have missed a 1/2 I am not sure:

View attachment 299894

Please look at the reference where you got this expression for $\nu$ and tell me what it says that it is. Is it the frequency of orbital motion of the electron when it is at state $n$ or is it something else?

Your calculation in part (a) is correct.

 

[drop_out_kid]

Please look at the reference where you got this expression for $\nu$ and tell me what it says that it is. Is it the frequency or orbital motion of the electron when it is at state $n$ or is it something else?

Your calculation in part (a) is correct.

It is the frequency, this is from my quantum physics class. I will give you a more detailed, I just can't figure out why I got different than the silde:

 

[drop_out_kid]

Please look at the reference where you got this expression for $\nu$ and tell me what it says that it is. Is it the frequency of orbital motion of the electron when it is at state $n$ or is it something else?

Your calculation in part (a) is correct.

So I cannot get agreed in (c):

And the bigger n is the bigger difference it gives:

 

[kuruman]

Please wait until I reply. We are not ready to move to part (c).

You have not told me how you interpret this $\nu$. Is it the frequency of orbital motion or not? The answer is important for part (c) and that is why I insist on you telling me what it is.

 

[drop_out_kid]

Please wait until I reply. We are not ready to move to part (c).

You have not told me how you interpret this $\nu$. Is it the frequency of orbital motion or not? The answer is important for part (c) and that is why I insist on you telling me what it is.

So from my understanding this should be orbital frequency instead of electron matter wave, this is before quantum mechanics part.

And I also pm you for some additional info..

 

[kuruman]

So from my understanding this should be orbital frequency instead of electron matter wave, this is before quantum mechanics part.

And I also pm you for some additional info..

I agree, it is not the frequency of a matter wave. If you look at the derivation you posted in #7, it is the frequency $\nu$ in $$h\nu=E_{\text{high}}-E_{\text{low}}$$ Does this suggest an orbit or something else?

 

[drop_out_kid]

I agree, it is not the frequency of a matter wave. If you look at the derivation you posted in #7, it is the frequency $\nu$ in $$h\nu=E_{\text{high}}-E_{\text{low}}$$ Does this suggest an orbit or something else?

I think this is for radiation of Atomic electron transition frequency, which is for part(b) (just my thought)

 

[berkeman]

And I also pm you for some additional info..

No, please do not do that, it is against the PF rules (see INFO at the top of the page). Please keep your requests for help in the open forums and not via PMs. Thank you.

 

[drop_out_kid]

No, please do not do that, it is against the PF rules (see INFO at the top of the page). Please keep your requests for help in the open forums and not via PMs. Thank you.

Ohh , sorry. It's not necessary anyway, please kindly ignore that message then.

So any idea how this problem work?

 

[berkeman]

So any idea how this problem work?

You are in very good hands with @kuruman

 

[kuruman]

I think this is for radiation of Atomic electron transition frequency, which is for part(b) (just my thought)

That's exactly what it is. You have the frequency of the orbiting electron $\nu_{\text{orb}}$ that you posted in #3 as $f$ and the frequency of radiation $\nu_{\text{rad}}$ that you posted in #4. As you noted, $\nu_{\text{rad}}$ is smaller by a factor of two. Your task in part (c) is to show that when $n$ becomes extremely large, $\nu_{\text{rad}}\approx \nu_{\text{orb}}.$ Got any any ideas on how to do that?

 

[drop_out_kid]

That's exactly what it is. You have the frequency of the orbiting electron $\nu_{\text{orb}}$ that you posted in #3 as $f$ and the frequency of radiation $\nu_{\text{rad}}$ that you posted in #4. As you noted, $\nu_{\text{rad}}$ is smaller by a factor of two. Your task in part (c) is to show that when $n$ becomes extremely large, $\nu_{\text{rad}}\approx \nu_{\text{orb}}.$ Got any any ideas on how to do that?

I think I actually did it. I mean I did it correct at first place but somehow I don't understand the result. Now it seems that , E=hf cannot be apply to (a) to verify the result, mvr=h_bar*n is very different than E=hf

 

[drop_out_kid]

That's exactly what it is. You have the frequency of the orbiting electron $\nu_{\text{orb}}$ that you posted in #3 as $f$ and the frequency of radiation $\nu_{\text{rad}}$ that you posted in #4. As you noted, $\nu_{\text{rad}}$ is smaller by a factor of two. Your task in part (c) is to show that when $n$ becomes extremely large, $\nu_{\text{rad}}\approx \nu_{\text{orb}}.$ Got any any ideas on how to do that?

They both in 1/n^3 degree terms. which by limit they are totally equal include the highest coefficients

 

[kuruman]

I think I actually did it. I mean I did it correct at first place but somehow I don't understand the result. Now it seems that , E=hf cannot be apply to (a) to verify the result, mvr=h_bar*n is very different than E=hf

They are expected to be different because mvr is angular momentum and E is energy. You need to compare apples with apples and oranges with oranges. What happened to the derivation of the orbital frequency that you posted? Please upload it using the "Attach files" on the lower left of the editor screen so that I can refer to it. I would also like to see the expression yo got for $\nu_{\text{rad}}$ when $n_2=n+1$ and $n_1=n$ in the limit $n\rightarrow \infty.$

 

[drop_out_kid]

They are expected to be different because mvr is angular momentum and E is energy. You need to compare apples with apples and oranges with oranges. What happened to the derivation of the orbital frequency that you posted? Please upload it using the "Attach files" on the lower left of the editor screen so that I can refer to it. I would also like to see the expression yo got for $\nu_{\text{rad}}$ when $n_2=n+1$ and $n_1=n$ in the limit $n\rightarrow \infty.$

Sure!
The limit is equal for sure.

So I am still a bit confused, let's say we get frequency in (a) right? that frequency is in 1/n^3

in (b) we use energy to get frequency and then it became 1/n^2??

So what's the difference of these two frequencies?

 

[kuruman]

The .pdf that you attached is sideways and I cannot read sideways. Please learn to post your work in LaTeX. The link is on the lower left of the editor screen, above the "Attach files button".

What you need to do here is to verify the Correspondence Principle. I suggest that you show that the semi-classical calculation for the change in energy when $\Delta n=1$ for large $n$ yields $\Delta E= h \nu_{\text{rad}}$. If you can show that $\nu_{\text{orb}}\approx \nu_{\text{rad}}$, then all you need to do is show that, semi-classically, when the angular momentum $L$ changes by $h$, the energy change is $\Delta E=h \nu_{\text{orb}}= \hbar \omega_{\text{orb}}.$ (I prefer $\omega$ over $\nu$.)

How to do that? I'll get you started. $$E=\frac{L^2}{mr^2}\implies~\Delta E=\frac{L\Delta L}{mr^2}.$$Assume $r$ does not change appreciably (it is proportional to $n$ which is huge) and find an expression for it in terms of $L$. What you need for $L$ and $\Delta L$ should be obvious.

 

[berkeman]

Here is the PDF with the rotation fixed. @drop_out_kid -- I agree with kuruman that you should learn to use LaTeX. It's a good skill in general, and it helps make your posts much more legible in discussion forums like PF.

 

[drop_out_kid]

The .pdf that you attached is sideways and I cannot read sideways. Please learn to post your work in LaTeX. The link is on the lower left of the editor screen, above the "Attach files button".

What you need to do here is to verify the Correspondence Principle. I suggest that you show that the semi-classical calculation for the change in energy when $\Delta n=1$ for large $n$ yields $\Delta E= h \nu_{\text{rad}}$. If you can show that $\nu_{\text{orb}}\approx \nu_{\text{rad}}$, then all you need to do is show that, semi-classically, when the angular momentum $L$ changes by $h$, the energy change is $\Delta E=h \nu_{\text{orb}}= \hbar \omega_{\text{orb}}.$ (I prefer $\omega$ over $\nu$.)

How to do that? I'll get you started. $$E=\frac{L^2}{mr^2}\implies~\Delta E=\frac{L\Delta L}{mr^2}.$$Assume $r$ does not change appreciably (it is proportional to $n$ which is huge) and find an expression for it in terms of $L$. What you need for $L$ and $\Delta L$ should be obvious.

Seems I did it in this way. Hope I did it right(there's a fixed pic of my pdf below)

 

[berkeman]

Seems I did it in this way. Hope I did it right(there's a fixed pic of my pdf below above)