Expanding the Commutator of 3 Operators A, B, and C - Quantum/Math Question

[cosmic_tears]

Quantum/math question :)

Homework Statement

I need to expand the commutator of the 3 operators A,B,C: [A,BC] in terms of [A,B], [A,C], [B,C].

Homework 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.

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.

 

[maverick_starstrider]

Are you sure you're supposed to get rid of the operators? I don't think it's possible to express it otherwise. That would require reducing a sum of 3 operators to a sum of 2 operators.

 

[cosmic_tears]

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?

 

[maverick_starstrider]

well an expansion of [A,BC] is ABC-BCA which is a sum of 3 operators in a row. I don't see how you could express that in the form aAB + bBA +cAC +dCA + eBC + fCB (where a...f are constants) which is the form it would need to take if you wanted it expressed solely in terms of [A,B], [A,C] and [B,C]

 

[cosmic_tears]

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!

 

[2Tesla]

Your answer will involve a sum of the products of the original operators with the two-operator commutators, as in:

X*[Y,Z] or [X,Y]*Z

Where X, Y, and Z are each one of A, B, and C. (I wrote it this way to avoid giving the answer away :)

 

[cosmic_tears]

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...

 

[2Tesla]

But clearly that's not possible. For example, let

A=p_{x}, B=x, C=y

Then

[A,BC] = [p_{x},x]*y = -i \hbar *y

and

[A,B] = -i\hbar, [A,C] = [B,C] = 0

There just isn't any way to do it. I think they mean to write it as an expression where the commutators involve only two of A, B, and C, like the one you gave above.

 

[cosmic_tears]

*sigh*... I'm really going to 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...