Conservation of Particle Number in the Nuclear Pairing Model

The Euler-Lagrange equations corresponding to a Bardeen-Cooper-Schrieffer state that is an eigenstate of the number operator are derived and solved numerically for a $\delta$ interaction. The errors due to the nonconservation of particle number in the usual Bardeen-Cooper-Schrieffer theory are studied as a function of particle number, level density, and strength of the pairing interaction. A proof is given that for attractive pairing interactions the lowest energy solution corresponds always to real positive probability amplitudes $v_{\nu }$, $u_{\nu }$.