# 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 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 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 at 17:46
• @ThomasFritsch I just copied the text from another post, sorry for not paying attention to that! – lawliet Mar 4 at 17:46