Energy splitting from spin-orbit interaction.

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SUMMARY

The discussion focuses on the energy splitting of the hydrogen 4f state due to spin-orbit interaction and the subsequent effects of a weak external magnetic field. Participants identify that the spin-orbit interaction leads to multiple energy states based on the alignment of angular momenta. The confusion arises regarding the distinction between energy splitting from spin-orbit interaction and the Zeeman effect, particularly in the context of weak magnetic fields. The need for a clear understanding of the perturbation term in the Hamiltonian for spin-orbit interaction is emphasized.

PREREQUISITES
  • Understanding of quantum mechanics principles, particularly angular momentum.
  • Familiarity with spin-orbit coupling concepts.
  • Knowledge of the Zeeman effect and its implications in quantum systems.
  • Basic proficiency in Hamiltonian mechanics and perturbation theory.
NEXT STEPS
  • Study the derivation of energy levels in hydrogen using spin-orbit coupling.
  • Learn about the perturbation term in the Hamiltonian for spin-orbit interaction.
  • Explore the differences between spin-orbit interaction and the Zeeman effect in detail.
  • Investigate the mathematical treatment of weak magnetic fields in quantum systems.
USEFUL FOR

Students and researchers in quantum mechanics, particularly those studying atomic physics and energy level splitting phenomena. This discussion is beneficial for anyone seeking to deepen their understanding of spin-orbit interaction and its effects on atomic states.

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Homework Statement


The spin-orbit interaction splits the hydrogen 4f state into many (a) Identify these states and rank them in order of increasing energy. (b) If a weak external magnetic field were now introduced (weak enough that it does not disturb the spin-orbit coupling), into how many different energies would each of these states be split?

Homework Equations


latex2png.2.php?z=100&eq=U%20%3D%20g_{Lande}\frac{e}{2m_e}m_j\hbar%20B_{ext}.jpg


latex2png.2.php?z=100&eq=B_{L}%20%3D%20\frac{\mu_0e}{4\pi%20m_er^3}L.jpg



The Attempt at a Solution


I feel like I understand how to do part b a bit better than part a. My book doesn't seem to be too helpful when dealing with energy splitting due solely to spin-orbit interaction, but goes into much greater detail about the Zeeman effect (presence of weak B field). In part a, I know energy will be split based on whether the angular momenta are aligned or not aligned, but I don't know how to find any split states besides that, and the book implies there are many. Then again, I don't understand why it should be any different than the Zeeman effect, wouldn't the B field perceived by the electron from the proton have the same effect as an external B field? I know the equations I posted give me a size for the difference in energy, but they don't seem to help me with finding the different states.
 
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What the perturbation term added to the Hamiltonian for the spin-orbit interaction?
 

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