# Gauge covariance of the magnetic momentum operator

In the book about Schrödinger Operators by Cycon et al. there is a step in their calculations I don’t understand. When I pick two vector potentials $$A_1$$ and $$A_2$$ such that their curl (i.e. the magnetic field) is the same then their difference must be the gradient of some smooth function (since the curl of the difference vanishes):

$$A_1 -A_2 = \nabla \lambda.$$

The conclusion is then the unitary equivalence of the magnetic Schrödinger operators $$(-i\lambda - A_j)^2$$, $$j=1,2$$. That is

$$e^{i\lambda}(-i\nabla - A_1)e^{-i\lambda} = (-i\nabla - A_2)$$

I’ve tried showing this by expanding the exponentials up to first order but got stuck. I’m sure this is an age old argument but I don’t see it right now.

Hint (recall the $$\nabla$$ is an operator acting to everything on the irght of it): $$-\text{i}\nabla e^{-\text{i}\lambda}= -\text{i}\left(\nabla e^{-\text{i}\lambda}\right) - e^{-\text{i}\lambda}\,\text{i}\nabla\,.$$