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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.

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

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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.

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  • $\begingroup$ 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
    Nov 4, 2019 at 23:41

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