Quantum/math question

Quantum/math question :)

1. The problem statement, all variables and given/known data
I need to expand the commutator of the 3 operators A,B,C: [A,BC] in terms of [A,B], [A,C], [B,C].

2. Relevant equations
[A,B] is defined to be AB - BA
also, [A,BC] = B[A,C] +[A,B]C.
There are some other identities but none that I see relevant.



3. The attempt at a solution

Tried lots of opening and additing/substructing of elements using the above formulas but I could never get rid of "operators". Is the a way to write "B" and "C" in terms of the above?
I'm working on this for over an hour! :(

Thank you very much!
Tomer.

 

Thanks for responding.

The question is:
Let A;B;C; be operators
1. Expand [A;BC] in terms of [A;B]; [A;C]; [B;C]

If you can understand anything different from the exact question you're most welcomed to explain to me :)
What do you mean by "That wuold require reducing a sum of 3 operators to a sum of 2"?
Why is that a consequence?

 

I still don't understand the "paradox" here, nor do I see how the sum you wrote (with the consts a...f) is a sun of two operators... what I see is 6 operators.

Ah well.... I thank you for trying anyway!

 

But I need an expression consisting only of [A,B], [A,C] and [B,C]
I already know that [A,BC] = B[A,C] + [A,B]C....

 

*sigh*... I'm really gonna hate them if you're right... :)
I'll send a mail to my tutor, all the student are too confused with this subject I guess none have noticed yet.

Thank you very much!
I'll post here if it turns out otherwise...