Lower energy levels with Dirac/Pauli theory than Schroedinger theory?

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Discussion Overview

The discussion centers around the comparison of energy levels calculated using the Dirac/Pauli equations versus those calculated with the Schrödinger equation, particularly in the context of the hydrogen atom. Participants explore the implications of relativistic effects on energy levels and the nature of these corrections.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that the lower energy levels calculated with the Dirac/Pauli equations may be related to relativistic effects and changing masses.
  • Others request specific examples to illustrate the claims, with a focus on the hydrogen atom.
  • One participant suggests that the relativistic corrections to kinetic energy are responsible for the observed lower energy levels, particularly for the first two levels of the hydrogen atom.
  • Another participant challenges the notion of relativistic mass, stating it is a deprecated concept and questioning its relevance to the discussion.
  • There is a suggestion that the energy corrections may not universally apply to all relativistic systems, indicating uncertainty about the generality of the claims.
  • Participants express confusion regarding the terminology, specifically the use of "Pauli equation" in this context, equating it to Schrödinger's equation with spin.
  • References to external sources, such as a Wikipedia article on fine structure, are provided to support claims.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the reasons for the lower energy levels or the implications of relativistic effects. Multiple competing views remain regarding the interpretation of energy corrections and the relevance of relativistic mass.

Contextual Notes

Some claims depend on specific examples, such as the hydrogen atom, and there is uncertainty regarding the applicability of relativistic corrections across different systems. The discussion also highlights potential misunderstandings about terminology and concepts related to relativistic physics.

Juli
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TL;DR
Lower energy levels with dirac/pauli theory than Schroedinger theory
Why do the enery levels calculated with the Dirac/Pauli euqations always lie lower than the results calculated with the Schrödinger equation?
I assume it has to do something with relativistic effects and the changing masses because of this.
 
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Please provide a specific example (or a reference to such an example)
 
hutchphd said:
Please provide a specific example (or a reference to such an example)
A specific example would be the hydrogen atom. I have linked a photo of what I mean.
Why are the corrected energys always lower? The relativistic parts are obviously taking up some energy and my question is where it goes. I don't think it's spin-orbin couling, since it just splits the levels up, some go higher, some lower. I think it's the kinetic and potential energy-terms. But why exactly? It probably has to do something with the relativistic mass, but aren't we using the rest mass in the Pauli-equation?

Screenshot 2025-01-17 160501.png
 
Juli said:
It probably has to do something with the relativistic mass, but aren't we using the rest mass

This effect should not be related to the relativistic mass. Relativistic mass is a deprecated concept that is no longer used to interpret special relativity, it is just a Lorentz factor in front of the mass.

Juli said:
A specific example would be the hydrogen atom. I have linked a photo of what I mean.
Why are the corrected energys always lower? The relativistic parts are obviously taking up some energy and my question is where it goes. I don't think it's spin-orbin couling, since it just splits the levels up, some go higher, some lower. I think it's the kinetic and potential energy-terms. But why exactly?

It is probably not true (as far as I know) that relativistic systems have lower energies than the non-relativistic ones. It is probably not even true for the hydrogen atom. As far as you have shown it is true for the first two levels of the hydrogen atom. In that case, yes it is due to the relativistic corrections to the kinetic energy as shown by the weakly relativistic calculation (at first order it's negative and larger than the other fine structure factors for the ground state and some of the first excited states).

Edit:

Juli said:
but aren't we using the rest mass in the Pauli-equation?

Also no idea what you mean by Pauli equation here, that's just Schrödinger's equation but with spin.
 
Last edited:
Juli said:
A specific example would be the hydrogen atom. I have linked a photo of what I mean.
Where does this photo come from? We need a reference.
 
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