Elliptical orbits in the Bohr-Sommerfeld model of the hydrogen. Five orbits are shown, with angular momentum 0, 1, 2, 3, and 4 in units of Planck's constant. The orbit of angular momentum zero does not go straight through the nucleus. https://en.wikipedia.org/wiki/Bohr%E2%80%93Sommerfeld_model#/media/File:Sommerfeld_ellipses.svg

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   John Carlos Baez (johncarlosbaez@mathstodon.xyz)'s status on Sunday, 09-Jul-2023 21:53:27 JST John Carlos Baez

   In the "old quantum theory", Bohr and Sommerfeld treated electrons as orbiting the nucleus along specific classical orbits whose action is an integer times Planck's constant. It was introduced in 1913. It ran into serious trouble in 1923, and it was later replaced by Schrödinger's "wave mechanics" and Heisenberg's "matrix mechanics", which turned out to be two ways of looking at the same theory.

   The old quantum theory is definitely wrong. But Manfred Bucher convincingly argues that it was abandoned for the wrong reason. The problems that killed it were caused not by its unavoidable deficiencies, but by a mental barrier - Sommerfeld's unwillingness to accept orbits where the electron goes straight through the atomic nucleus!

   It's obvious that if you seek an orbit where the angular momentum is zero, it must go straight through the nucleus. But look at this picture on Wikipedia: the orbit called ℓ = 0, which should have zero angular momentum, is a narrow ellipse that does *not* have zero angular momentum!

   YES, it seems crazy to have the electron go straight through the nucleus. That's why Sommerfeld said these orbits were "unphysical" and dropped them.

   But if you allow them, the predictions of the old quantum mechanics are improved. The problems that killed it - the failure of Bohr, Born, Pauli and Heisenberg to calculate the right ground state energy for helium, Pauli's failure to find a stable ground state for the ionized hydrogen molecule H₂⁺, and finally Pauli's failure to calculate how a hydrogen atom reacts to combined electric and magnetic fields - largely go away!

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