Help - Verify the Jacobi Identity (Arfken)

[Fjolvar]

Hello, I'm unfamiliar with the notation used in this problem with the commas. I understand matricies, identities, etc. but not sure about the commas..

Question 3.2.9: Verify the Jacobi Identity: [A,[B,C]] = [B,[A,C]] - [C,[A,B]]

I see the BAC CAB rule here, but not sure how to show it. Any help on this problem would be greatly appreciated!

 

[vela]

The square brackets denote the commutator: [A,B] = AB-BA.

 

[Fjolvar]

Any suggestions on setting this problem up to prove it? Just make 2x2 matricies with a1,a2,a3,a4 b1,b2,b3,b4.. etc? I haven't a clue.

 

[vela]

Just expand both sides and show you end up with the same terms.

 

[Fjolvar]

I know this should be easier than I'm making it. I tried to expand these in many ways but couldn't get it to work out. Since AB does not equal BA for example since we're dealing with matrices, I don't know how to prove this algebraically.. How do I expand this?

 

[vela]

Post what you tried so we can see what you're doing.

 

[Fjolvar]

Ok I attached what I have so far. In the first attempt I tried expanding everything out with matrices and then realized I actually left out a step in each part which would make things very messy so I thought there had to be an easier way, and in the second attempt I tried without using matrices but stopped because I'm not sure how to distribute since AB does not equal BA..

 

[vela]

Your final attempt is the way to go. All you have to do is finish off the righthand side.

You found, with sign corrections, that

[B,[A,C]]=BAC-BCA-ACB+CAB
[C,[A,B]]=CAB-CBA-ABC+BAC

so

[B,[A,C]]-[C,[A,B]] = (BAC-BCA-ACB+CAB)-(CAB-CBA-ABC+BAC)

Now just simplify it as some of the terms cancel, and you'll be left with what you have for (1).

 

[Fjolvar]

Sometimes I think I just need more confidence in doing these problems. Thank you very much!