I know that the Bohr model of the atom was modified by Sommerfeld to include elliptical orbits, and that the resulting Bohr-Sommerfeld theory has long been put aside in favor of quantum mechanics.

Nevertheless, I wanted to ask if there has ever been a more general formulation of the Bohr-Sommerfeld theory (either recently or in the past) through which one could quantize the paths that electrons can take in any desired atomic/molecular system, ultimately to arrive at certain geometries and electron trajectories in 3D space that correspond to the $n$, $l$, and $m_l$ quantum numbers?

I ask this question because I have a working idea of how to generalize the Bohr-Sommerfeld theory to possibly any system, and wanted to know if it would be feasible to publish my idea. By this I mean I actually wrote a java program to try and implement this idea computationally. I'm just trying to get some results for different choice atomic/molecular systems and publish them, hopefully under some guidance since I have never published a paper before.

Edit: I would also like to briefly mention a paper I've recently read that attempted to correct some of the inaccuracies in Sommerfeld's expansion of Bohr's model. I plan to incorporate those corrections into my calculations as well. The paper is titled "Rise and premature fall of the old quantum theory," written by Manfred Bucher in 2008 (https://arxiv.org/abs/physics/0605258). Of note, the corrections made involve removing the circular orbit from the model and introducing a linear, coulomb oscillation orbit through the hydrogen atom's nucleus to give the correct l = 0, 1, 2, ..., n-1 values for the orbital angular momentum quantum number. It also includes the correction that the angular momentum of the hydrogen atom is proportional to $\sqrt{l(l+1)}$, not $l$ as Sommerfeld originally proposed.

  • $\begingroup$ There is a book by L. J. Curtis, "Atomic Structure and Lifetimes: A Conceptual Approach" about semiclassical atomic physics. $\endgroup$
    – user137289
    Commented Nov 4, 2019 at 22:25
  • $\begingroup$ Thanks for the suggested reading. I will take a look. $\endgroup$
    – mathTrials
    Commented Nov 4, 2019 at 23:51

1 Answer 1


Well I don't know how your new theory will work but the problem that you will have to face, is that the Bohr-Sommerfeld quantization only works for problems that are separable. By that I mean that $p_i$ can only depend on $q_i$ and not on any $q_j$ with $j \neq i$. This forbids the calculation of Atoms with more than one electron.

I am curious how you are planning to solve that problem.

  • $\begingroup$ If you simulate the particles in a system based on coulomb forces of attraction & repulsion, you can see how the momentum $p_i$ of one particle (such as an electron) is affected by the location $q_j$ of another particle (such as other electrons & atomic nuclei). My java program runs this simulation based on the initial positions and momenta that are assigned to each particle in a given system. $\endgroup$
    – mathTrials
    Commented Nov 4, 2019 at 23:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.