Projection of angular momentum via linear algebra

Projection of many-body states with good angular momentum from an initial state is usually accomplished by a three-dimensional integral. We show how projection can instead be done by solving a straightforward system of linear equations. We demonstrate the method and give sample applications to ${}^{48}\mathrm{Cr}$ and ${}^{60}\mathrm{Fe}$ in the $pf$ shell. This new projection scheme, which is competitive against the standard numerical quadrature, should also be applicable to other quantum numbers such as isospin and particle number.

Figure 1

Fraction of given angular momentum $J$ ($f_J$) in a Slater determinant, calculated both for ordinary Hartree-Fock (HF) and cranked HF with cranking frequency $\hslash \omega =1.0$ MeV, for ${}^{60}\mathrm{Fe}$ in the $pf$ shell. The maximum angular momentum for this nuclide in this model space is 26 $\hslash$.

Figure 2

Yrast excitation energies for ${}^{60}\mathrm{Fe}$ computed in the $pf$ shell with the semiphenomenological interaction gxpf1a. Results are from full shell-model diagonalization (SM, black circles) and linear algebra projected Hartree-Fock (PHF, red squares). For comparison we also show the experimental values (blue diamonds). $Y$ axis scaled as $J(J+1)$.