Stern-Gerlach Experiment: Spin Orbit Coupling

[eep]

Hi,
We're looking at the consequences of the Stern-Gerlach experiment which has lead us to the idea of spin orbit coupling in the hydrogen atom. In class, the coupling term of the Hamiltonian was derived by assuming the electron has a magnetic moment, then viewing the hydrogen atom in the electron's frame. We then said that the proton orbits the electron, which is effectively a current, thereby producing a magnetic field which interacts with the moment. Obviously, this idea of the proton having a circular trajectory is totally crap in quantum mechanics, but everyone I've asked hasn't been able to give me a different derivation. All I've gotten is that this is one of those cases where classical mechanics gives a result which happens to work in the quantum case. Anyone know how to get the result without this idea of an orbiting proton?

 

[jtbell]

It looks like you have to start with the (relativistic) Dirac equation instead of the (non-relativistic) Schrödinger equation, and solve it for the hydrogen atom. In doing so, you have to introduce the electric and magnetic fields together by way of the four-vector potential which includes both the electric scalar potential and the magnetic vector potential.

http://zopyros.ccqc.uga.edu/lec_top/rltvt/node5.html

This automatically gives you the effects of spin-orbit coupling in the energy levels. If you then take the non-relativistic limit of the Dirac equation, you get the Schrödinger equation with a correction term which is the same as the one that you derived using the "orbiting proton" picture.

http://zopyros.ccqc.uga.edu/lec_top/rltvt/node6.html

 

[denian]

Hello, as a scientist, I would like to address your question regarding the derivation of the spin orbit coupling term in the hydrogen atom without the assumption of an orbiting proton. Firstly, I would like to clarify that the concept of an orbiting proton in the hydrogen atom is a classical interpretation and does not accurately represent the quantum mechanical behavior of the atom.

The spin orbit coupling term arises from the interaction between the intrinsic angular momentum (spin) of the electron and its orbital angular momentum. This interaction can be described by the Dirac equation, which is a relativistic equation that accurately describes the behavior of the electron in the hydrogen atom.

To derive the spin orbit coupling term, one can start with the Dirac equation and use perturbation theory to expand the Hamiltonian to first order. This approach takes into account the relativistic effects and results in the spin orbit coupling term without the need for assuming an orbiting proton.

Furthermore, the spin orbit coupling term can also be understood from a quantum mechanical perspective by considering the electron’s wave function as a superposition of different states with different orbital angular momenta. This results in an effective magnetic field that interacts with the electron’s spin, giving rise to the spin orbit coupling term.

In summary, the spin orbit coupling term in the hydrogen atom can be derived without the classical assumption of an orbiting proton. The concept of an orbiting proton is a useful analogy but does not accurately represent the quantum mechanical behavior of the atom. I hope this explanation helps to clarify your doubts.