Quantizing the Toda lattice

In this work we study the quantum Toda lattice, developing the asymptotic Bethe ansatz method first used by Sutherland. Despite its known limitations we find, on comparing with Gutzwiller's exact method, that it works well in this particular problem and in fact becomes exact as ℏ grows large. We calculate ground state and excitation energies for finite-sized lattices, identify excitations as phonons and solitons on the basis of their quantum numbers, and find their dispersions. These are similar to the classical dispersions for small ℏ, and remain similar all the way up to ℏ=1, but then deviate substantially as we go farther into the quantum regime. On comparing the sound velocities for various ℏ obtained thus with that predicted by conformal theory we conclude that the Bethe ansatz gives the energies per particle accurate to O(1/${\mathrm{N}}^2$). On that assumption we can find correlation functions. Thus the Bethe ansatz method can be used to yield much more than the thermodynamic properties which previous authors have calculated.