The Magnus expansion and the Schrieffer-Wolff transformation are both methods used to get certain effective Hamiltonians. I know that at a basic level, the Schrieffer-Wolff transformation eliminates high-energy interactions, and the Magnus expansion is a series of nested commutators which, if the series is taken to infinity, equals the original Hamiltonian.
What are the differences between these methods of transforming the Hamiltonian? Say I have a Hamiltonian and I take a Magnus expansion of it and also transform it by Schrieffer-Wolff. I know that I should be able to describe the same physics with both methods, but will the terms be exactly the same?
