What Is the Name of the Commutator Relation [A,exp(X*B)]?

[Nusc]

Homework Statement

[A,exp(X*B)] = exp(X*B)[A,B]X

Is there a name for this relation?

Homework Equations

The Attempt at a Solution

If not, how do you prove it?

A(X*B)^n/n! - (X*B)^n/n! * A

 

[Nusc]

Nevermind.

 

[Nusc]

Actually,

http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/CoherentStates.pdf

how does [A,B^2] = 2B[A,B]?

Also,

how is <1|x|0> nonzero?

doesn't position operators anhilate the vacuum state?