However, its development took quite a long time. The Bethe ansatz for particles with short-ranged interactions, can be sketched in the following way. In one-dimensional systems particles can only exchange their position by scattering with each other, due to the special topology. Therefore one can split the wave function in sectors given by the order of the particles. In particular one can restrict the calculation of the wave function in the sector with All other sectors are given by permutations of this sector.
Particle moves freely as long as .
Therefore one can assume a plain wave in this region. There remains the problem of adjusting the boundaries of the plain waves. It is therefore hard to find a model, where the Bethe ansatz can be applied. The Quantum Inverse Scattering Method provides the possibility to search systematically for Bethe ansatz solvable models, i.e. it allows the construction of models which can be solved exactly using the Bethe ansatz.