Pauli spin matrices

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SUMMARY

The discussion centers on the mathematical representation of the dot product of Pauli spin matrices, specifically the identity πœŽπ‘–β‹…πœŽπ‘—=πœŽπ‘–π‘₯πœŽπ‘—π‘₯ + πœŽπ‘–π‘¦πœŽπ‘—π‘¦ + πœŽπ‘–π‘§πœŽπ‘—π‘§. Participants explore expressing this identity purely in terms of πœŽπ‘§, noting that the x and y components alter spin states. The use of ladder operators, defined as ##\sigma_{\pm}=\sigma_x\pm i\sigma_y##, is emphasized for this transformation. Additionally, the implications for the proton spin wavefunction are discussed, highlighting the relevance of these mathematical tools in quantum mechanics.

PREREQUISITES
  • Understanding of Pauli spin matrices
  • Familiarity with quantum mechanics concepts
  • Knowledge of ladder operators in quantum theory
  • Basic grasp of spin wavefunctions
NEXT STEPS
  • Research the application of ladder operators in quantum mechanics
  • Study the role of Pauli matrices in quantum state transformations
  • Explore the implications of spin states on particle physics
  • Investigate the mathematical representation of spin wavefunctions
USEFUL FOR

Physicists, quantum mechanics students, and researchers focusing on spin systems and particle physics will benefit from this discussion.

Eliena
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TL;DR
I tried to solve this expression (Οƒi β‹…Οƒj)Οƒk in terms of zth component of pauli spin matrices. So that i can apply the spin projection operator to the baryonic wavefunctions. As the Οƒy component contains the imaginary terms, so i think it is more convenient to write spin in terms of Οƒz only. Please give suggestions.
Using the identity for the dot product of Pauli matrices: πœŽπ‘–β‹…πœŽπ‘—=πœŽπ‘–π‘₯πœŽπ‘—π‘₯ + πœŽπ‘–π‘¦πœŽπ‘—π‘¦ + πœŽπ‘–π‘§πœŽπ‘—π‘§. How to express this purely in terms of πœŽπ‘§, as x and 𝑦 components change spin states
 
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You can write it using ladder operators ##\sigma_{\pm}=\sigma_x\pm i\sigma_y##.
 
pines-demon said:
You can write it using ladder operators ##\sigma_{\pm}=\sigma_x\pm i\sigma_y##.
How it could be applied to the proton spin wavefunction?
 

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