  Every solution of the Bethe-ansatz equations (BAE) is characterized by a set
of quantum numbers, by which we can evaluate it numerically. However, no
general rule is known how to give quantum numbers for the physical solutions of
BAE. For the spin-1/2 XXX chain we rigorously derive all the quantum numbers
for the complete set of the Bethe-ansatz eigenvectors in the two down-spin
sector with any chain length $N$. Here we obtain them both for real and complex
solutions. Consequently, we prove the completeness of the Bethe ansatz and give
an exact expression for the number of real solutions which correspond to
collapsed bound-state solutions (i.e., 2-string solutions) in the sector: $2[
(N-1)/2 - (N/\pi) \tan^{-1}(\sqrt{N-1})]$ in terms of Gauss' symbol. Moreover,
we prove in the sector the scheme conjectured by Takahashi for solving BAE
systematically. We also suggest that by applying the present method we can
derive the quantum numbers for the spin-1/2 XXZ chain.
