The Schrieffer–Wolff transformation is a unitary transformation which expresses the Hamiltonian in a basis (the "dressed" basis) where it is diagonal to first order in the perturbation V . This unitary transformation is conventionally written as: H ′ = e S H e − − S When V is small, the generator S of the transformation will likewise be small. The transformation can then be expanded in S using the Baker-Campbell-Haussdorf formula H ′ = H + [ S , H ] + 1 2 [ S , [ S , H ] ] + … … Here, [ A , B ] is the commutator between operators A and B . In terms of H 0 and V , the transformation becomes H ′ = H 0 + V + [ S , H 0 ] + [ S , V ] + 1 2 [ S , [ S , H 0 ] ] + 1 2 [ S , [ S , V ] ] + … … The Hamiltonian can be made diagonal to first order in V by choosing the generator S such that V + [ S , H 0 ] = 0 This equation always has a definite solution under the assumption that V is off-diagonal in the eigenbasis of H 0 . Substituting this choice in the previous transformation yields: H ′ = H 0 + 1 2 [ S , V ] + O ( V 3 ) This expression is the standard form of the Schrieffer–Wolff transformation. Note that all the operators on the right-hand side are now expressed in a new basis "dressed" by the interaction V to first order.