The Bethe Ansatz

Let us start again from the \(N=2\) case, which illustrates the general construction. The periodicity conditions can be written as

\begin{equation*} \Psi_2 (x_1 + L, x_2| \lambda_1, \lambda_2) = \Psi_2 (x_1, x_2 + L| \lambda_1, \lambda_2) = \Psi_2 (x_1, x_2| \lambda_1, \lambda_2). \end{equation*}

Looking at l.psi2 and invoking the linear independence of the free waves with distinct rapidities gives the following quantization conditions for the rapidities:

\begin{equation*} e^{i\lambda_1 L} = - e^{-i \phi (\lambda_1, \lambda_2)}, \hspace{1cm} e^{i \lambda_2 L} = - e^{i \phi(\lambda_1, \lambda_2)}, \end{equation*}

which can also be rewritten using l.phi as

\begin{equation*} e^{i \lambda_1 L} = \frac{\lambda_1 - \lambda_2 + ic}{\lambda_1 - \lambda_2 - ic}, \hspace{10mm} e^{i \lambda_2 L} = \frac{\lambda_2 - \lambda_1 + ic}{\lambda_2 - \lambda_1 - ic}. \end{equation*}

These are known as Bethe equations. In view of the state classification, their most convenient form is obtained by taking the logarithm,

\begin{equation*} \lambda_1 + \frac{1}{L} \phi (\lambda_1 - \lambda_2) = \frac{2\pi}{L} I_1, \hspace{1cm} \lambda_2 + \frac{1}{L} \phi (\lambda_2 - \lambda_1) = \frac{2\pi}{L} I_2, \end{equation*}

where

\begin{equation*} I_1, I_2 \in \mathbb{Z} + \frac{1}{2} \end{equation*}

are half-odd integers uniquely labelling the quasimomenta \(\lambda_1, \lambda_2\), and thus take the role of quantum numbers of the theory. Subtracting the two equations,

\begin{equation*} \lambda_1 - \lambda_2 + \frac{2}{L}\phi (\lambda_1 - \lambda_2) = \frac{2\pi}{L} (I_1 - I_2), \end{equation*}

and using the monotonicity of the phase shift function l.phi shows that the sign of \(I_1 - I_2\) is the same as that of \(\lambda_1 - \lambda_2\). Therefore, if \(I_1 = I_2\), we must have \(\lambda_1 = \lambda_2\). In this case however, as we have seen, the wavefunction identically vanishes, manifesting the Pauli principle for the Bethe Ansatz.

The total momentum and energy of this two-particle wavefunction are simply

\begin{equation*} P = \lambda_1 + \lambda_2, \hspace{10mm} E = \lambda_1^2 + \lambda_2^2. \end{equation*}