- #1

- 195

- 5

## Homework Statement

Given that the function f can be expanded in a power series of [itex]a[/itex] and [itex]a^\dagger[/itex], show that:

[itex][a,f(a,a^{\dagger})]=\frac{\partial f }{\partial a^\dagger}[/itex]

and that

[itex][a,e^{-\alpha a^\dagger a}] = (e^{-\alpha}-1)e^{-\alpha a^{\dagger} a}a[/itex]

## The Attempt at a Solution

I've tied using simple algebra, but got nowhere. Writing the exponential operators as infinite sums did not seems to be leading to an equality either. I am not sure how to correctly order the result of the derivative resulting from the first equation. I.e. [itex]e^{-\alpha a^{\dagger} a}(-\alpha a)[/itex] or [itex](-\alpha a)e^{-\alpha a^{\dagger} a}[/itex].