Undergrad K dot p method with SO interaction

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SUMMARY

The discussion focuses on deriving the k dot p formula while incorporating the spin-orbit interaction term. The user is encountering challenges with the treatment of the cross product term resulting from the spin-orbit interaction and the implications of the Pauli matrix vector on the spinor. Additionally, the user seeks clarification on how to apply the momentum operator p to Bloch states, particularly in the context of degenerate perturbation theory and diagonalizing the sigma matrices.

PREREQUISITES
  • Understanding of k dot p perturbation theory
  • Familiarity with spin-orbit coupling in quantum mechanics
  • Knowledge of Bloch wave functions and their properties
  • Proficiency in working with Pauli matrices and spinors
NEXT STEPS
  • Study the derivation of the k dot p method with spin-orbit interaction
  • Learn about the application of Pauli matrices in quantum mechanics
  • Research degenerate perturbation theory and its implications
  • Explore the mathematical treatment of Bloch states under momentum operators
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Quantum physicists, researchers in condensed matter physics, and students studying semiconductor physics who are interested in advanced topics related to spin-orbit coupling and perturbation theory.

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I am trying to derive the k dot p formula with the spin orbit interaction term included. I am having some trouble. I insert a bloch wave function multiplied with an unspecified spinor. But how do I treat the cross product term that comes from the SO and what is the effect of the pauli matrix vector on the spinor? And also: Since the spin orbit term contain the gradient of the coulomb potential how can I let p act on the bloch states? (I assume this how you derive the correct formula?)
The result I want is the attached. Also I attached what I have written up so far.
 

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This is a problem of degenerate perturbation theory. I would first try to diagonalize the sigma matrices.
 
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