Commutation relation of operators

[mathfilip]

Im reading in a quantum mechanics book and need help to show the following relationship, (please show all the steps):

If A,B,C are operators:

[A,BC] = B[A,C] + [A,B]C

 

[stevenb]

Im reading in a quantum mechanics book and need help to show the following relationship, (please show all the steps):

If A,B,C are operators:

[A,BC] = B[A,C] + [A,B]C

Just expand out the commutation operators based on the definition, i.e. [X,Y]=XY-YX.

When you see how easy this is, you will laugh and might even be embarrassed that you posted such an easy question - no offense intended ... I've done much worse.

 

[ismaili]

Just expand out the commutation operators based on the definition, i.e. [X,Y]=XY-YX.

When you see how easy this is, you will laugh and might even be embarrassed that you posted such an easy question - no offense intended ... I've done much worse.

I've often done much worse too.
Moreover, there are some similar formulas, which can be proven easily and similarly:
$$[A,BC] = \{A,B\} C - B\{ A , C \}$$
$$\{ A , BC \} = \{A,B\}C - B[A,C] = [A,B]C + B\{A,C\}$$
where the square bracket denotes the commutator and the curly bracket denotes the anti-commutator.

Probably mathfilip wanted to specify the point that the algebra must be "associative", or these formulas are not valid.