Undergrad Pauli matrices and shared eigenvectors

[Sunny Singh]

TL;DR

For a spin 1/2 particle, why does Sx, Sy and Sz don't share the complete eigenspace even though all of them commute with S^2

We know that S² commutes with S_z and so they share their eigenspace. Now since S² also commutes with S_x, as per my understanding, the eigenvectors of S² and S_z should also be the eigenvectors of S_x. But since the paulic matrices σ_x and σ_y are not diagonlized in the eigenbasis of S², it is clear that S² and S_x don't share their eigenspace even though they commute with each other. How is that possible? what am i missing?

 

[fresh_42]

To have common eigenspaces we need that all commute with each other. Every matrix commutes with $I$, but that doesn't mean all matrices have the same eigenspace. And the $S_{xyz}$ do not commute.

 

[PeterDonis]

We know that S2 commutes with Sz and so they share their eigenspace.

Yes.

since S2 also commutes with Sx, as per my understanding, the eigenvectors of S2 and Sz should also be the eigenvectors of Sx.

Your understanding is incorrect. S2 and Sx share an eigenspace, but it's a different eigenspace from the one shared by S2 and Sz. The two eigenspaces must be different because Sx does not commute with Sz.

 

[Truecrimson]

To say it in another way, commutativity is not transitive. $[A,B] = 0$ and $[B,C]=0$ does not imply that $[A,C]=0$.