# Why does the Bohr model give the correct energy levels for hydrogen even though it assumes a circular orbit?

I've been reading the answers for this question: Why did the Bohr Model Successfully calculate some of the energy levels in hydrogen? but it hasn't really satisfied my curiosity. Some answers suggest it's luck and it cannot be derived from some more accurate theory (like GR -> Newton's gravitational law or Lorentz transformation -> Galileo transformation).

I've read this comment which I don't fully understand:

Take Schrodinger's equation Set $$\psi =a e^{i\frac{J}{\hbar}}$$ in. One will get an equation for $$a$$ and for $$J$$ higher order terms of $$\hbar$$ like $$\hbar^2$$ are to be neglected. Then you get the Hamilton-Jacobi-equation (HJE) With the HJE you can compute the Bohr's model. So one kind of develops Schrodinger's equation in orders of $$\hbar$$ and one gets a fundamental classical mechanics equation which can be used for getting Bohr's model (adding the quantisation principle).

Does someone understand this?

Also, I'm just really puzzled how something correct can come out of e.g. considering the electron orbit to be circular which is assumed to be wrong for multiple reasons.

• – Qmechanic Mar 4 '20 at 16:41
• What does 𝜓=𝑎𝑒𝑖𝐽ℏ mean? May be $\psi=ae^{iJ/\hbar}$ ? For better readability please use MathJax in your question. – Thomas Fritsch Mar 4 '20 at 16:42
• @Qmechanic I don't fully see the connection, do I need to consider symmetries? I still don't understand how assuming something very wrong (definite circular orbit) results in correct energy levels. – lawliet Mar 4 '20 at 17:46
• @ThomasFritsch I just copied the text from another post, sorry for not paying attention to that! – lawliet Mar 4 '20 at 17:46

## 1 Answer

Nothing about the Bohr model of the atom is really correct. There are no orbits. So circular or elliptical is a moot point. The reason the Bohr model predicts some things well is just coincidence. You need quantum mechanics to correctly explain the hydrogen atom.

• It was built from knowledge of the Balmer series, so it involved fitting the orbit sizes to that empirical data. It was only later that Debroglie came up with the wavelength argument for the choice of orbit sizes, and indirectly that lead to the idea of a wave, which lead to a wave equation, which lead to Schroedinger's work on producing his quantum wave equation. Bohr's approach could have been fitted to any data as long as it was closed under addition and subtraction. At the time it was criticised as being numerology. – Ponder Stibbons Mar 5 '20 at 1:32
• Yes but that doesn't in any way make it correct. – Natsfan Mar 5 '20 at 1:35
• In fact, I was saying the Bohr model was junk. Just overfitting a model to the data. And it was only out there for a couple of years before it got heavily modified and then replaced entirely. No reason it should even be brought up in quantum mechanics courses. – Ponder Stibbons Mar 5 '20 at 1:37
• Okay - partly I was saying that it was not a coincidence that it worked, as far as that it was a model that could be fitted to any spectral series. To me, this means as physics it was junk. It told us nothing about the atom. My understanding is that we are on the same page with that. – Ponder Stibbons Mar 5 '20 at 1:38