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

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Homework Help Overview

The discussion revolves around combining commutators in quantum mechanics, specifically related to a problem involving momentum operators and spin matrices. The original poster is attempting to solve a question from a quantum mechanics assignment that involves the manipulation of these operators.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • The original poster discusses their approach to the problem, expressing uncertainty about the commutation relations between momentum and position operators, as well as the treatment of the sigma matrices. They question the validity of their steps and seek clarification on the logical reasoning behind their manipulations.

Discussion Status

Participants are actively engaging with the problem, with some expressing confusion and seeking further explanation. There is a recognition of the need for a logical framework to understand the steps taken, particularly regarding the commutation relations and the treatment of the operators involved. No consensus has been reached yet.

Contextual Notes

The original poster mentions potential assumptions about the commutation of the sigma matrices with linear operators and the use of an alternative representation for the Hamiltonian. There is also a concern about the validity of results derived in earlier parts of the problem.

FunkyDwarf
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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
 
Last edited by a moderator:
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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
 
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!
 
ok I am really stuck guys, pwease help? =D
 

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