## Problem

Let and be linear operators, which commute with the commutator of and , i.e.

Show that the following formulae hold:

which are known as the *Baker-Campbell-Hausdorff identities*. They play an important rule, e.g. in quantum mechanics, whenever one deals with exponentiation of the position and the momentum operator.

## Solution

We will start by proving the first equation. To this end we introduce an auxiliary operator , which depends on a real parameter :

We will show, that , which will prove the claim. Clearly this statement is true for . The derivative of with respect to is

It is easy to see by induction, that under the conditions imposed on and ,

which implies

and thus and consequently validates the first identity.

In order to prove the second equation, we will do a similar trick and define

Obviously, , and the derivative

In the second step I have interchanged with in the third term. The commutation rest cancels the fourth term.

This completes the proof.

I remember I had to prove this theorem in a “Quantum Mechanics II” course homework exercise a few years ago, and it took forever to find clues for the solution online, until I found out some page where this was proven, but it was called “Glauber’s formula” there.