Realistic Microscopic Calculations of Nuclear Structure

Abstract

One would like to start with the free nucleon-nucleon (NN) interaction and many-body quantum mechanics and solve for the properties of finite nuclei. In principle, this simply involves solving the many-body Schrödinger equation:

$$H|{{\Psi }_{\alpha }}\rangle = {{E}_{\alpha }}|{{\Psi }_{\alpha }}\rangle$$

(1)

for the eigenenergies E _α and the eigenstates $|{{\Psi }_{\alpha }}\rangle$ of the many-particle system, where a is some label characterizing the states. But it is impossible to solve this problem in the full Hilbert space S when the number of particles in the system exceeds a certain limit because it contains too many degrees of freedom. Consequently, one wishes to truncate the problem to a smaller space S of dimension d, in which it becomes tractable to carry out the calculation. Now let $|{{\Phi }_{\beta }}\rangle$ represent the projections of d of the states $|{{\Phi }_{\beta }}\rangle$ into S. Thus we define the effective Hamiltonian H in S to satisfy

$$\mathcal{H}|{{\Phi }_{\beta }}\rangle = {{E}_{\beta }}|{{\Phi }_{\beta }}\rangle ,$$

(2)

where the eigenvalues {E _β } are dof the exact eigenvalues {E _α } in Eq.(1). Because the $|{{\Phi }_{\beta }}\rangle$ are projections of the $|{{\Psi }_{\alpha }}\rangle$, they are, in general not orthogonal.