A Pauli spin matrices

[Eliena]

TL;DR Summary

I tried to solve this expression (σi ⋅σj)σk in terms of zth component of pauli spin matrices. So that i can apply the spin projection operator to the baryonic wavefunctions. As the σy component contains the imaginary terms, so i think it is more convenient to write spin in terms of σz only. Please give suggestions.

Using the identity for the dot product of Pauli matrices: 𝜎𝑖⋅𝜎𝑗=𝜎𝑖𝑥𝜎𝑗𝑥 + 𝜎𝑖𝑦𝜎𝑗𝑦 + 𝜎𝑖𝑧𝜎𝑗𝑧. How to express this purely in terms of 𝜎𝑧, as x and 𝑦 components change spin states

 

[pines-demon]

You can write it using ladder operators $\sigma_{\pm}=\sigma_x\pm i\sigma_y$.

 

[Eliena]

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?