Two-Body Equations for Four-Nucleon Problems

Abstract

The variational principle for the energy is used to derive integrodifferential equations for the two-body functions which, when combined in the product form, yield the "best" independent-pair wave function for the a particle. A practicable iteration procedure for finding approximate solutions to these equations is developed and is used to obtain an approximate ground-state wave function, for an example, four-bodyHamiltonian for which the potential is central. Improvements in the iteration procedure are described which make it a feasible method for solving the integrodifferential equations even when tensor forces are included.

Bounding the ground-state energy of a many-body system with the differential method

“…For M = 3 we have already seen that α 3 = 1/4; for d = 3, (27) reproduces exactly (23). No better estimate is obtained when considering M = 4.…”

“…Crossed numerics and analytical studies lead to very plausible conjectures on the geometrical description of the configuration for M = 5 and M = 8 for which explicit analytical value of α M can be proposed [16]. The lower bound for (27), for N M , is strictly improved when increasing M from 5 and in particular is better than (23).…”

“…(12)]. First, when N is not too large for the numerical computation to remain tractable, the direct calculation of sup QN F N shows (see figure 1) that it gives better lower estimates than (23). For very large N , we can nevertheless benefit from the maximum of F M for smaller M .…”

“…Each solid line starting at M corresponds to the lower bounds (27). For M = 3 and M = 4, there is only one solid curve given by (23). The dashed line corresponds to the upper bound (20) given by [13, eq.…”

“…When the potential V has the form (2), a natural choice of trial function is to take (for variational techniques in a few nuclear body context, such a choice has been used by [18,23] for instance)…”