Extended hamiltonian operator for the Hydrogen atom

[PedroBittar]

I am familiar with the usual solution of the hydrogen atom using the associated legendre functions and spherical harmonics, but my question is: is it possible to extend the hamiltonian of the hydrogen atom to naturally encompass half integer spin?
My guess is that spin only pops in naturally in relativistic quantum mechanics, but since my knowledge on the subject is limited I am not sure how to proceed with the argument and would appreciate some references with more complete discussions.

 

[blue_leaf77]

My guess is that spin only pops in naturally in relativistic quantum mechanics

Your guess hits the point. If you stay non-relativistic, electron spin never enters the Hamiltonian of a Hydrogen-like atom (there is a perturbation method to account for the effect of spin, but as you might know a perturbation is something you add into the original model, it's not something you start off from the beginning). In fact, the relativistic effect in atom is best described through Dirac equation. For a more detailed discussion you may try chapter 8 in "Modern Quantum Mechanics" 2nd edition by Sakurai and Napolitano.

 

[dextercioby]

I am familiar with the usual solution of the hydrogen atom using the associated legendre functions and spherical harmonics, but my question is: is it possible to extend the hamiltonian of the hydrogen atom to naturally encompass half integer spin? [...]

Yes, it is possible to extend the Hamiltonian by the Levy-Leblond approach.

[...] My guess is that spin only pops in naturally in relativistic quantum mechanics, but since my knowledge on the subject is limited I am not sure how to proceed with the argument and would appreciate some references with more complete discussions.

Your guess is wrong. Levy-Leblond's work is the subject of chapter 13 of Walter Greiner's "Quantum Mechanics. An Introduction", Springer Verlag.
Actually, Pauli's equation is the simplest Galilei-group invariant wave equation, just in the same vein as the Dirac one is the simplest Poincaré-group invariant group equation. There were/are two famous Ukrainian physicists, Fushchich and Nikitin who wrote a whole book on group-theoretical derivations of wave equations

 

[PedroBittar]

Yes, it is possible to extend the Hamiltonian by the Levy-Leblond approach.
Your guess is wrong. Levy-Leblond's work is the subject of chapter 13 of Walter Greiner's "Quantum Mechanics. An Introduction", Springer Verlag.
Actually, Pauli's equation is the simplest Galilei-group invariant wave equation, just in the same vein as the Dirac one is the simplest Poincaré-group invariant group equation. There were/are two famous Ukrainian physicists, Fushchich and Nikitin who wrote a whole book on group-theoretical derivations of wave equations

View attachment 114767

That is very interesting, and makes a lot of sense since the pauli equation seems to be the nonrelativistic limit of dirac equation.
Thanks, I will definitely take a look on this group theory derivations.