How do you combine commutators in quantum mechanics for problem Q1?

[FunkyDwarf]

Hey guys,

Tryin to do Q1 in http://members.iinet.net.au/~housewrk/QM/AQM2006.ex.newnotation.pdf and I am having trouble in b.) i get the commutator equal to
c * permutation tensor (sigma . p * (xi pk) -xi pk * sigma . p) and i know I am missing some cruical step to recombine this, ie i assume the momentum operator and the sigma matrices commute but the position and momentum operators dont? furthermore i can't see how to substitute in for the position operator, do we use the alternative representation for H as i hbar d/dt ?

Hope that made sense ><
Cheers
-G

 

[FunkyDwarf]

Ok if i expand the dot product as a sum over j indicies (i thought it would have to be say over l because its an independent sum so shouldn't have anything to do with epsilon but working backwards from the question...) and magically compress that down using [p,x] = - i hbar i can sort of do it...but there's a lot of magical handwaving in that =P can someone provide a logical explanation as to why that works? (if it works) I also assumed i can pull the sigma matrices out front which seems fair enough as they should commute with linear operators, i think...(and hope)

CHeers
-G

 

[FunkyDwarf]

Hmm also for part e i get that the ith component of the spin operator is equal to the negative ith component of the spin operator? that seems wrong...also to get that i didnt use part d which makes it seem more wrong...help!

 

[FunkyDwarf]

ok I am really stuck guys, pwease help? =D