Given a linear operator![Mathworld](http://mathworld.wolfram.com/favicon_mathworld.png)
, we define:
It follows that
|
|
|
(2) |
Consider another linear operator . Let . Then one can prove the following series representation for :
where and .
A very important special case of eq. (3) is known as the
Baker-Campbell-Hausdorff (BCH) formula. Namely, for we get:
|
|
|
(4) |
Alternatively, this expression may be rewritten as
|
|
|
(5) |
or
|
|
|
(6) |
There is a descendent of the BCH formula, which often is also referred to as BCH
formula. It provides us with the multiplication law for two exponentials![Planetmath](http://planetmath.org/sites/default/files/fab-favicon.ico)
of linear operators: Suppose . Then,
|
|
|
(7) |
Thus, if we want to commute two exponentials, we get an extra factor