Prove that the Bohr hydrogen atom approaches classical conditions when [..]

[s3a]

"Prove that the Bohr hydrogen atom approaches classical conditions when [. . .]"

Homework Statement

The problem and its solution are attached as ProblemSolution.jpg.

Homework Equations

E_k = chR/(n_k)^2
E_l = chR/(n_l)^2
ΔE = hc/λ
hc/λ = chR[1/(n_k)^2 – 1/(n_l)^2]
1/ λ = R[1/(n_k)^2 – 1/(n_l)^2]

The Attempt at a Solution

Given E_k = chR/(n_k)^2 and E_l = chR/(n_l)^2,

ΔE = chR[1/(n_k)^2 – 1/(n_l)^2]

Therefore, since, ΔE = hc/λ,

hc/λ = chR[1/(n_k)^2 – 1/(n_l)^2]
1/ λ = R[1/(n_k)^2 – 1/(n_l)^2]
1/ λ = R[1/(n_k)^2 – 1/(n_l)^2]
1/ λ = R[1/(n_k)^2 – 1/(n_l)^2
ν = R[(n_l)^2 – (n_k)^2]/[(n_k)^2 * (n_l)^2]

and, n_l – n_k = -1 which counters the negative that I had initially compared to the answer of the book so far and now the only difference is that my answer lacks the c multiplicative factor that the book has. If I did something wrong, what is it? Or is it the book?

Also, how is the “crazier” part of equation (1.6.3) obtained?

If more information is needed, just ask.

Any help would be greatly appreciated!
Thanks in advance!

 

[TSny]

my answer lacks the c multiplicative factor that the book has. If I did something wrong, what is it? Or is it the book?

What is the relationship between frequency, wavelength, and c?

Also, how is the “crazier” part of equation (1.6.3) obtained?

Just substitute the well-known Bohr model expressions for the radius of the orbit and the speed of the electron in the orbit.

 

[s3a]

1) The relationship between frequency, wavelength, and c is v = c/λ.

2) I found the equation for the radius “on a silver-platter” and derived the equation for the velocity and yes, plugging them in worked.

3) Now, I still have some work which disagrees with the solution and it is attached as MyWork.jpg. Could you please tell me if I am wrong or if it's the solution that is wrong as well as what as what to do to get the correct answer if I am wrong?

 

[TSny]

S3a, The energy levels are negative: E = -chR/n². The text that you quoted in your original post left out the minus sign for some reason (misprint?)

 

[s3a]

Could it be that the negative sign is to indicate that the (light) energy is being lost from the Hydrogen orbital whereas here we are talking about the energy gained by the photon?

I deduced this based on the following quote from the screen-shot of the problem and its solution that I attached initially.: "Therefore the frequency of the emitted photon is [. . .]". The fact that that equation is the frequency of the emitted photon should imply that the frequency multiplied by h (plank's constant) is the energy of the emitted photon gained (rather than that lost from the Hydrogen orbital).

That would explain why the book has it the way it does but not why I get the book's answer multiplied by -1. Is there something with my logic above that doesn't hold?

 

[TSny]

To me the natural way to think about it is that the energy of the photon equals the loss of energy of the atom as the atom goes from the higher excited state (n_k) to the lower excited state (n_l).

Thus h\nu = E_k - E_l where E_k = -chR/n_k² and E_l = -chR/n_l²

 

[TSny]

Long before the Bohr model was developed, it was known that the frequencies of hydrogen could be calculated by taking the difference of numbers of the form cR/n² where n is a positive integer and R was an empirically determined number. These numbers cR/n² were called "terms".

The Bohr model later explained this numerology by showing that the energy levels of hydrogen were just the negative of these terms multiplied by h and the model derived the value of R in terms of fundamental constants.

Maybe your book is using these "terms" rather than energy levels of the atom.

 

[s3a]

What you said earlier makes sense but could it also be the case that it just doesn't matter what the sign is because what the question asks for ultimately is frequency and we just want to compare the magnitudes of the two frequencies (classical and modern) so we take the absolute value? Is that a valid thought process?

 

[TSny]

What you said earlier makes sense but could it also be the case that it just doesn't matter what the sign is because what the question asks for ultimately is frequency and we just want to compare the magnitudes of the two frequencies (classical and modern) so we take the absolute value? Is that a valid thought process?

That sounds good to me.

 

[s3a]

Okay, thanks for all your help!