Effective hamiltonians

Abstract

Non-linear equations, generalizing those determining Bloch's transformation to an effective hamiltonian, are set up in a very simple way. It is shown how they immediately determine the results of Van Vleck, Bloch and Soliverez. Treating the Foldy-Wouthuysen transformation as a special case of the Van Vleck transformation, sixth-order expressions are obtained by means of the non-linear equations. The transformations of Soliverez, Roussy and Kvasnicka are shown to be strictly identical—because they all have a simple property which is proven to be unique for that of Soliverez. Kvasnicka relates his transformation to that of Primas'. We shall see that it is also very closely related to that of Van Vleck's. A certain non-hermitian effective hamiltonian also given by Roussy is proven to be identical to that of Bloch's. Primas' level shift transformation gives an effective hamiltonian for each level in zeroth order. This effective hamiltonian is seen to be identical to that of Soliverez's up to and including third order—and to all orders if one makes a simple change in ‘normalization’. Computer treatment of finite matrices is discussed. The sixth-order Van Vleck hamiltonian recently given by Jørgensen and Pedersen in terms of commutators is derived in a more simple way.