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

[Juli]

TL;DR Summary

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.

 

[hutchphd]

Please provide a specific example (or a reference to such an example)

 

[Juli]

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?

 

[pines-demon]

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.

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:

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.

 

[PeterDonis]

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.

 

[Juli]

Where does this photo come from? We need a reference.

I got it from the wikipedia article about fine structure https://en.wikipedia.org/wiki/Fine_structure