I Pauli matrices and shared eigenvectors

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

fresh_42

Mentor
Insights Author
2018 Award
10,429
7,122
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.
 
24,895
6,249
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.
 
262
86
To say it in another way, commutativity is not transitive. ##[A,B] = 0## and ##[B,C]=0## does not imply that ##[A,C]=0##.
 

Want to reply to this thread?

"Pauli matrices and shared eigenvectors" You must log in or register to reply here.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Hot Threads

Top