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While pursuing forth in Atomic Structure, I encountered the following concepts...

According to Bohr, the angular momentum of an orbit remains quantized, i.e $\displaystyle mvr = n\hbar .$

But he wasn't able to expalin the reason that why is it so.

But later Louis de Broglie gave his hypothesis:

Every particle in the universe has dual nature, one wave nature an other particle nature.

Focusing on wave nature of particle he gave an equation, known as de Broglie equation or de Broglie wavelength as

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

This was proved years later by Davisson and Germer's Diffraction Experiment.

Now, we can also prove the quantization of angular momentum of an orbit as:

For any orbit the number of waves produced by an electron in one complete revolution is $\displaystyle \frac {2\pi r}{\lambda}.$

Substituting the values of $r,v$ given by Bohr, then

$$\displaystyle \frac {2\pi r}{\lambda} = \frac {2\pi r \cdot mv}{h} = \frac { n^2 \hbar^2 \cdot mkZe^2}{\hbar \cdot mkZe^2 \cdot n\hbar} = n$$

As $\displaystyle r =\frac { n^2 \hbar^2 }{ mkZe^2} $ and $\displaystyle v = \frac {kZe^2}{ n\hbar}.$

Now , as $ \displaystyle \frac {2\pi r}{\lambda}= n ,$ From this we can prove that angualr momentum of an orbit is quantized.

Now according to Quantum wave mechanical model, the angular orbital momentum of an orbital is : $ \displaystyle \sqrt {l(l+1)} \hbar .$

Now which one is actually valid, because in many of the books both of the momentums are refered only by the term angular momentum. But both are not same.

My question might sound silly, for such topic... But I would like to know which one is to be refered when we encounter angular momentum.

Nikola Alfredi
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  • Seriously, erase the Bohr model from your brain. It is scientifically wrong. Its only use is giving us a glimpse into how quantum mechanics developed historically. – orthocresol Apr 26 '20 at 11:10
  • Ok, means we have to consider Quantum wave mechanical model whenever angular momentum is refered... Thank you. – Nikola Alfredi Apr 26 '20 at 11:12
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    Yep. The Bohr model did manage to get some things right, but those were essentially by coincidence. Electrons simply don't move in a circular "orbit" around the nucleus. – orthocresol Apr 26 '20 at 11:13
  • @orthocresol what is the significance of angular momentum in quantum model when electron does not got in circular motion? – Vishal prabhu lawande Aug 18 '20 at 17:06
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    @Vishalprabhulawande, even classically, angular momentum is simply the cross product of position and momentum: $\mathbf{l} = \mathbf{r} \times \mathbf{p}$. It is not something that is exclusive to uniform circular motion. The same is true of quantum angular momentum, except that $\mathbf{r}$ and $\mathbf{p}$ are operators. – orthocresol Aug 18 '20 at 17:10
  • @orthocresol got it. thank you so much! Any good reading matter you can suggest in understanding quantum model of an atom?... Would be a great help – Vishal prabhu lawande Aug 18 '20 at 17:30
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    @Vishalprabhulawande My suggestion is to read an actual QM book, and by that I mean a physics one. In my opinion, (notwithstanding the famous Feynman quote) one needs to "understand" QM itself before they can "understand" QM as applied to atoms. I learnt from Griffiths' Introduction to Quantum Mechanics, but there are many texts out there. You don't need to follow it through to the end with all the advanced topics, but getting the basics right is really important. – orthocresol Aug 18 '20 at 17:58

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Bohr Model came before the quantum theory and it was a success for explaining the non- continuous emission spectrum of excited hydrogen atoms. However, Bohr model was not sufficient for explaining many other phenomena related to the interaction of matter with light, such as the non-continuous emission spectrum of multi-electronic atoms, Zeeman Effect,... Therefore, the quantum theory is considered as a more complete theory. The quantification of the angular orbital momentum should be linked to the second quantum number $l$ not $n$.

Up till now, we use Bohr model in chemistry textbooks in spite of its limitation. This is because it's simple and doesn't require complex mathematical tools as quantum theory.

Yomen Atassi
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