In considering the many body problem of the nucleus, the independent particle model is often used as a starting point. In the independent particle model (IPM) the nucleons in the nucleus are taken to be non-interacting (uncorrelated) particles moving in some mean field potential. An occupation of the lowest single-particle states up to, and not beyond, the Fermi energy describes the ground state. This model has been successful in describing the shell structure of the nucleus and in describing the characteristics of single particle excitations, but in order to describe collective phenomena one needs to also consider the residual two-body interactions between nucleons which are not accounted for in the mean-field potential. It is these residual interactions which give rise to collectivity in the nucleus. Excitations of the ground state are described in terms of particle-hole excitations in which a nucleon in some state, k, below the Fermi surface is promoted to a state, m, above it. The Hamiltonian of these excitations is given by:
where and a are the creation and annihilation operators, respectively, is the ground state energy of the nucleon, and is the particle-hole interaction.
If the independent particle model does a fairly good job of describing the ground state nucleus then the Random Phase Approximation (RPA), which cuts off the excitations at the one-particle/one-hole (1p-1h) level, will give a good description of the nuclear excitation. The 1p-1h excitations in RPA can be formally defined by the operator
The index m(i) indicates a state below(above) the fermi surface. and are the forward and backward-going p-h amplitudes, respectively.
Physically the squares of these amplitudes represent the probability of finding a given p-h excitation in the final state,
The equation of motion for ,
after linearization can be written as [Ost92]
which is the standard RPA equation. The matrix elements are given by
It is obvious that a calculation of the excitation spectrum is highly dependent on the residual p-h interaction. Generally, RPA provides a valid description of nuclear excitations providing that the backward-going amplitudes, , are small compared to the forward-going amplitudes, . If the former are large it is an indication that the ground state will deviate significantly from the independent particle model due to 2p-2h and higher order excitations. In other words the residual interaction is causing large ground state correlations.