# Behaviour of an Exponential Commutation

1. Mar 31, 2013

### d.hatch75

1. The problem statement, all variables and given/known data
Forgive the awkward title, it was hard to think how to describe the problem in such a short space. I'm following the proof of Proposition 1.2 of Nakahara's "Geometry, Topology and Physics", and the following commutation relation has been established:

$\partial_x^n e^{ikx} = e^{ikx}(ik + \partial_x )^n$

The line immediately afterwards is:

$e^{-i\epsilon \{-\partial_x^2 / 2m + V(x)\}}e^{ikx} = e^{ikx}e^{-i\epsilon \{-(ik+\partial_x )^2 / 2m + V(x)\}}$

I am not seeing how this follows from the previous relation, since the partial derivative is in the exponential so surely it doesn't act on $e^{ikx}$ in the same way? It seems dodgy, but I can only assume I'm missing some important result concerning exponentials of commutation relationships that hasn't otherwise been specified anywhere in the book as far as I can tell.

2. Relevant equations
(See above)

3. The attempt at a solution
(See above)

2. Apr 1, 2013

### BruceW

hint: try using the power series expansion of an exponential function.