Cyclic permutation and operators

[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)$

 

[Sourabh N]

They are just referring to the short hand notation: [S_i,S_j] = S_iS_j - S_jS_i = 2i S_k 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.

 

[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.

 

[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?

 

[MRAH]

Yes thanks a lot, I appreciate your help.