# Cyclic permutation and operators

1. Oct 10, 2012

### MRAH

Hi there

I am working through the problems in R.I.G. Hughes book the structure and interpretation of quantum mechanics and have hit a wall in the last part of the following question:

Show that Sx and Sy do not commute, and evaluate SxSy-SySx. Express this difference in terms of Sz, and show that this relation holds cyclically among the three operators.

I guess it has something to do with cyclic permutation. Any way thanks for your time and if you know where I can find the answers to the problems in this book that would help me later I suppose.

S$_{}$x= 1/2 $\left($0 1
10$\right)$ S$_{}$y= 1/2 $\left($0 -i
i 0$\right)$ S$_{}$z= 1/2 $\left($1 0
0 -1$\right)$

Last edited: Oct 10, 2012
2. Oct 11, 2012

### Sourabh N

They are just referring to the short hand notation: [Si,Sj] = SiSj - SjSi = 2i Sk where $(i,j,k)$ can be $(x,y,z)$ or $(y,z,x)$ or $(z,x,y)$, hence the phrase "this relation holds cyclically among the three operators".

S represent the pauli spin matrices.

Last edited: Oct 11, 2012
3. Oct 11, 2012

### MRAH

My confusion is with the section in bold type and how exactly to show the relation holding cyclically among the operators. What I was trying to depict below was the Pauli spin matrices.

4. Oct 11, 2012

### Sourabh N

By "show that this relation holds cyclically among the three operators" they mean what I have written in my earlier post. I edited the commutation relation (in boldface) to comply with Pauli spin matrices. Made a few other changes to explain it better. Is that helpful?

5. Oct 11, 2012

### MRAH

Yes thanks a lot, I appreciate your help.