Comparisons of BCS Nuclear Wave Functions with Exact Solutions

The validity of the Bardeen-Cooper-Schrieffer variational method is re-examined for two nuclear-model cases, one resembling the system of deformed nuclei and the other spherical nuclei. For simplicity, a constant-pairing-force approximation is adopted. The projected BCS wave functions compare badly with the exact wave functions at pairing-force strength around the critical force strength, and only for sufficiently strong pairing forces does the BCS method seem to approach the exact. For the spherical case, the average value of the pair operator ($\frac{1}{2}N$) is also calculated over a wide range of force strength, and the general behavior is found to be consistent with the results obtained on the wave-function components. The errors in eigenvalues are computed, and discussions are given as to the possible sources of the deviations.