Isospin Invariance and the Pairing-Force Problem

Abstract

The isoscalar pairing-force Hamiltonian is studied first in the BCS-like approximation, including neutron-neutron, proton-proton, and neutron-proton interactions. Constraining $〈\hat{N}〉$ and $〈{\hat{T}}^2〉$, we find a class of BCS-like wave functions which give the same minimum energy; they differ in the expectation value of ${\hat{T}}_z$, $-T<~〈{\hat{T}}_z〉<~T$. For the limit $〈{\hat{T}}_z〉=T$, the minimum occurs at zero neutron-proton interaction. The residual interactions, those neglected by the BCS approximation, are treated by the quasiboson approximation. The spurious effects of number and isospin dispersion are identified. A procedure for explicitly displaying the number and isospin dependence of the energy is given, together with one for obtaining excited states with all possible isospins. Finally, as an example, the degenerate case is worked out, and agreement with the exact solution, including ground-state neutron-proton correlations, to the order considered, is demonstrated.