The Bethe Ansatz

Many particles c.l.s.N

The above considerations for two particles readily generalize to the case of an arbitrary number of particles. One defines a fundamental open domain

\begin{equation*} D_N: x_1 < x_2 < ... < x_N \hspace{10mm} \mbox{with boundaries} \hspace{10mm} \partial_j D_N: x_{j+1} = x_j + 0^+, \end{equation*}

the solution of the time-independent Schrödinger equation l.se1 being equivalent to the solution of the system

\begin{equation*} \left( H_{\small LL}^{\small (N)} - E_N \right) \Psi_N ({\bf x}) \bigr|_{{\bf x} \in D_N} = 0, \hspace{10mm} \left(\partial_{x_{j+1}} - \partial_{x_j} - c \right) \Psi_N ({\bf x}) \bigr|_{{\bf x} \in \partial_j D_N} = 0. \end{equation*}

If we consider the region of \(\mathbb{R}^N\) in which at most two coordinates exactly coincide, the full \(N\)-body wavefunction is given by a straightforward generalization of the two-body result l.psi2,

\begin{align} \Psi_N({\bf x} | {\boldsymbol \lambda}) &= \prod_{N \geq j_1 > j_2 \geq 1} sgn(x_{j_1} - x_{j_2}) sgn (\lambda_{j_1} - \lambda_{j_2}) \times \nonumber \\ & \times \sum_{P \in \pi_N} (-1)^{[P]} e^{i \sum_{j=1}^N \lambda_{P_j} x_j + \frac{i}{2} \sum_{N \geq j_1 > j_2 \geq 1} sgn(x_{j_1} - x_{j_2}) \phi (\lambda_{P_{j_1}} - \lambda_{P_{j_2}})}. \tag{l.psin}\label{l.psin} \end{align}

The Bethe wavefunction thus has four distinctive features:

  • it is a mixture of plane waves,
  • each plane waves carrying a rapidity (internal quasi-momentum) label,
  • these plane waves are combined using relative amplitudes set by microscopic two-body interactions
  • the internal quasi-momenta obey a form of Pauli exclusion (since the wavefunction identically vanishes in the case of pairwise equal rapidities, as direct inspection reveals).

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Author: Jean-Sébastien Caux

Created: 2024-01-18 Thu 14:24