Generator coordinate method with a conjugate momentum: Application to particle number projection

We discuss an extension of the generator coordinate method (GCM) by taking simultaneously a collective coordinate and its conjugate momentum as generator coordinates. To this end, we follow the idea of the dynamical GCM (DGCM) proposed by Goeke and Reinhard. We first show that the DGCM method can be regarded as an extension of the double projection method for the center of mass motion. As an application of DGCM, we then investigate the particle number projection, for which we not only carry out an integral over the gauge angle as in the usual particle number projection but also take a linear superposition of Barden-Cooper-Schrieffer (BCS) states which have different mean particle numbers. We show that the ground state energy is significantly lowered by such effect, especially for magic nuclei for which the pairing gap is zero in the BCS approximation. This suggests that the present method makes a good alternative to the VAP method, as the method is much simpler than the VAP.

Figure 1

The probability to find the $N_0=28$ component in the BCS wave function $|N\rangle$ for ${}^{56}\mathrm{Ni}$ which has the average neutron number of $N$.

Figure 2

The energy gain in the ${}^{56}\mathrm{Ni}$ nucleus as a function of the number of basis states $N_{\mathrm{DGCM}}$ in the DGCM method. It is plotted with respect to the energy of $N_{\mathrm{DGCM}}=1$, which is equivalent to the variation before projection method. The solid, the dashed, and the dotted lines denote the results with ${\lambda }_{\mathrm{cut}}={10}^{-4},{10}^{-5}$, and ${10}^{-6}$ for the cut-off of the eigenvalues of the overlap kernel, respectively.

Figure 3

Similar to Fig. 2, but for a comparison between the total energy (the solid lines) and the pairing energy (the dashed lines). The filled circles and the filled triangles denote the results for the ${}^{56}\mathrm{Ni}$ and the ${}^{58}\mathrm{Ni}$ nuclei, respectively.

Figure 4

The pairing gap $\mathrm{\Delta }$ of ${}^{56}\mathrm{Ni}$ in the BCS approximation for the basis states for the DGCM calculation.