- #1
yoyopizza
- 39
- 1
- Homework Statement
- I'm working on problem 1.10 in Sakurai (1.8 in version 1 or 2), where I'm supposed to prove the commutation relation on [S_i,S_j]=i\epsilon_{ijk}h_bar*S_k
- Relevant Equations
- S_x=hbar/2(|+><-|+|-><+|)
S_y=ihbar/2(-|+><-|+|-><+|)
S_z=hbar/2(|+><+|-|-><-|)
Trying to use <+|+>=1=<-|-> and <-|+>=0 to prove each iteration of the equation, so I have 6 different versions to prove. But the part I'm currently stuck on is understanding how to simplify any given version. I've written out [S_x,S_y]=S_xS_y\psi-S_yS_x\psi and expanded it in terms of the |+>,|-> kets and bras. Then using associativity I've broken up some of the three term components like <-|+|-> into (<-|)(+|->) which I'm assuming will allow me to remove some terms, however now I don't really understand what + or - means in the absence of being inside a ket or bra. I can't imagine they equal their usual matrix representation because then +=|+> which makes no sense.