1

I use the information from the text titled The Bohr Atom written by Michael Fowler and the Wikipedia page titled Larmor formula while framing my question.

In the section titled "The Correspondence Principle" in the former text, Dr. Fowler writes, "Or, more simply, if it's going round the circle at frequency $f$ revolutions per second, it will be emitting radiation at that frequency $f$ —because its electric field, as seen from some fixed point a few meters or so away, say, will be rotating $f$ times per second." when he talks about the energy radiated by the electron revolving around the proton in the "large" H-atom.

However, the Larmor formula says that the power of the radiation emitted by an accelerating charged particle is proportional to the square of its acceleration. i.e. $$P \propto a^2$$$$\Rightarrow P \propto {\omega}^4$$$$\Rightarrow P \propto f^4$$ The formula gives that the power radiated is proportional to the frequency to the 4th power. What is the problem with this approach? Is it that we can not use it for circular motion? Or is it something else?

Also, I do not understand why for a large orbit, $$E_{n + 1} - E_{n} = hf$$ Firstly, because the right hand side of the equation represents continuous radiation due to the revolving electron, and the left hand side represents one transfer of electron between two states, so how can they be equated?

And secondly, why do we only choose two adjacent levels, since I do not think it should matter as $n$ is taken to be really large, so the difference in the values of $n$ should not make a difference.

I think that is all from me, thank you for your time and effort.

Qmechanic
  • 220,844
Aditya Bansal
  • 21
  • 6

2 Answers2

1

Niels Bohr quantized the atom by treating the problem via the classical method of action angle variables, or Hamilton-Jacobi theory. The correspondence principle provided Bohr with the justification needed to extend a method of classical dynamics to a problem that is non-classical, i.e. quantum. The quantization of the electron's angular momentum is a result of Hamilton-Jacobi theory to the hydrogen atom under the assumption that the action $J$, of the electron for a complete cycle of motion is discrete or quantized: $$J=\oint L\; dt=nh.$$ This is known as Bohr's quantum condition, and is not derived but assumed as an hypothesis. Similarly, Plank and Einstein made quantum assumptions in their treatments of the blackbody (discrete electromagnetic modes) and photoelectric (discrete energy packets) problems respectively.

Albertus Magnus
  • 5,652
0

Bohr derived the angular momentum quantization mainly by expressing kinetic energy in terms of angular momentum. He did not use the Correspondence Principle for that.

He presented the Correspondence Principle much later to imply that quantum mechanics 'approximates' to classical mechanics beyond a certain range.

What made Bohr quantise angular momentum and not some other quantity?

Lalit Patel
  • 381