A Pauli spin matrices

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The discussion focuses on expressing the dot product of Pauli matrices, specifically πœŽπ‘–β‹…πœŽπ‘—, purely in terms of πœŽπ‘§, emphasizing that the x and y components alter spin states. It highlights the use of ladder operators, defined as ##\sigma_{\pm}=\sigma_x\pm i\sigma_y##, to facilitate this expression. Participants explore the implications of this formulation for understanding proton spin wavefunctions. The conversation underscores the mathematical relationships between the Pauli matrices and their application in quantum mechanics. Overall, the discussion aims to deepen the understanding of spin states through the lens of Pauli matrix identities.
Eliena
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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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