Hydrogen Degeneracy: Qualitative Explanation

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Discussion Overview

The discussion centers on the degeneracy of hydrogen energy levels, specifically why the orbital angular momentum quantum number l does not influence energy levels in hydrogen-like atoms. Participants explore related questions about the impact of additional protons and electron configurations on this degeneracy.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant notes that hydrogen energy levels depend solely on the principal quantum number n, questioning the role of the orbital angular momentum quantum number l.
  • Another participant explains that the 1/r nature of the Coulomb potential leads to the degeneracy of states with the same n but different l, referring to this as "accidental degeneracy."
  • A participant mentions that the essential degeneracy of 2l+1 states with the same l but different m is due to the rotational invariance of central potentials.
  • Some participants suggest that atoms with one electron maintain this degeneracy, while those with full shells plus one electron may not due to electron shielding affecting the potential.
  • One participant introduces the concept of a conserved quantity related to symmetry, referencing the quantum Noether theorem as a basis for the accidental degeneracy.
  • Another participant expresses interest in the connection between classical mechanics and quantum mechanics, specifically mentioning the Lenz vector.
  • One participant recalls seeing this topic in lecture notes but notes a lack of discussion in standard textbooks.

Areas of Agreement / Disagreement

Participants express varying views on whether hydrogen-like atoms maintain degeneracy and how electron configurations influence this property. The discussion remains unresolved regarding the implications of additional protons and electron shielding on degeneracy.

Contextual Notes

Participants reference the 1/r potential and its implications for degeneracy without fully resolving the mathematical details or assumptions involved in their arguments.

Allday
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The hydrogen energy levels (when only considering the coulomb field of an infinitely massive proton - or when using the reduced mass for the electron) only depend on the principle quantum number n. Can someone give me a qualitative argument why the orbital angular momentum quantum number l doesn't come into play here?

related questions:

do hydrogen like atoms (ionized to the point of having only one electron but containing more than one proton) break this degeneracy between the different orbital angular momentum states?

do atoms with the same number of protons and electrons, but with one electron outside of completely full shells break this degeneracy?

gonna read some and see if i can find the answer
 
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It turns out the 1/r nature of the coulomb potential is the reason why all the states with the same n and different l have the same energy. The degeneracy is called accidental to differentiate it from the essential degeneracy of the 2l+1 states with the same l but different m. This essential degeneracy is due to the rotational invariance of any central potential.

This leads me to believe that the atoms with one electron maintain the degeneracy, while those with full shells + 1 don't because the electron shielding will alter the 1/r potential.
 
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The "accidental" degeneracy is due to a conserved quantity corresponding to a symmetry of the system via quantum Noether theorem.

See https://faculty.washington.edu/seattle/physics324/Lenz.pdf for details.
 
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thanks for the link to the paper dextercioby. I remember learning about that Lenz vector when I was doing classical mechanics orbits. Interesting to see its quantum equivalent in play. I think it would be nice if they included a discussion of this in more of the standard quantum books.
 
I've seen this issue treated in some lecture notes before, but i can't remember any textbook author discussing it.
 

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