# Excitation operators and intrinsic Hamiltonians

## Abstract

An operator A† that satisfies [H, A†] = ℏωA† converts a stationary-state eigenfunction of the Hamiltonian H into another eigenfunction with energy eigenvalue increased by ℏω. Such operators describe collective excitations of many-particle systems, and their properties can be used to construct an intrinsic Hamiltonian that is dynamically independent of the collective degrees of freedom, without introducing subsidiary conditions. The procedure developed by Lipkin, valid when ℏω is real and positive, is extended to make possible the construction of an intrinsic Hamiltonian when ℏω vanishes and A† is Hermitian, and also when ℏω is complex. The nuclear cranking model is shown to be a special case of the proposed general method for vanishing ℏω in which effective moments of inertia occur as eigenvalues of linear equations. Several examples are worked put in detail, all dealing with an interacting phonon-electron system in the random-phase approximation. Results derived are the explicit screened Coulomb interaction resulting from electronic plasma excitations, a verification of the renormalized phonon frequency spectrum and phonon-electron interaction derived in the adiabatic approximation, and the resulting screened Coulomb and phonon-induced electronic interactions obtained when plasma and phonon excitations are treated simultaneously.