Electron wave energies different for +½ and -½ spin?

[Garlic]

Main Question or Discussion Point

Hello everyone,
How come the electron wave energy states are different for spin -½ and +½ values? What causes this asymmetry?

 

[vanhees71]

I don't know, what's "wave energy". Do you mean the energy-eigenvalue problem (time-independent Schrödinger eqaution)? Than it's clear that for electrons in a magnetic field the energy eigenvalues depend on the spin component in direction of the magnetic field, because the spin is associated with a magnetic moment.

 

[Garlic]

I was referring to the energy states of an electron that is orbiting an atom. I have red somewhere (I can't find the source at the moment), that the potential energy of the orbiting electron is different for different spin values, even when other quantum numbers have the same values (n,m,l).

 

[vanhees71]

Yes it is, because there's a spin with electron(s) implying a magnetic moment. In the momentary rest frame of the electron the nucleus moves and thus provides a magnetic field, leading to spin-orbit coupling. In this argument you have take into account that the transformation between the momentary electron rest frame and the rest frame of the nucleus must be done with Lorentz rather than Gailei transformations, which leads to the gyromagnetic factor of 2 (Thomas precession); or you use the Dirac rather than the non-relativistic Pauli equation right away, where the spin-orbit coupling comes out right without any cumbersome Lorentz transformations, because the Dirac equation is relativistically covariant. These corrections (together with the other relativistic corrections on top of the non-relativistic Pauli equation) to the atomic spectra are called fine structure of the spectra:

https://en.wikipedia.org/wiki/Fine_structure#Spin-orbit_coupling

Then you also have hyperfine structure, which is due to the interaction of the nucleus's magnetic moment with the magnetic field due to the electrons.

https://en.wikipedia.org/wiki/Hyperfine_structure

Last but not least there are radiative corrections taking into account the quantum nature of the electromagnetic field. For the hydrogen atom that's known as the Lamb shift:

https://en.wikipedia.org/wiki/Lamb_shift

 

[Garlic]

Thank you!