Excited States Configurations of the Quantum Toda Lattice

Abstract

Excited states configurations of the quantum Toda lattice are studied by the direct diagonalization of the Hamiltonian. The most probable configurations of one-hole and one-particle excitations are shown to be similar to the profiles of classical phonon and soliton excitations, respectively. One-hole excitation states, which are always ground states of definite Em-symmetry of the dihedral group DN, change those structures abruptly with the potential range varied. One-particle excitations, which are buried in complicated excitation spectra, have well-defined configurations similar to the cnoidal profile of the classical periodic Toda lattice. The relationship that the hole (particle) excitations in quantum mechanics correspond to the phonon (soliton) excitations in classical mechanics, which has been suggested based on the similarity of dispersion relations, is confirmed in a geometrically understandable way. Based on the study of one-soliton and two-soliton states, the structure of multi-soliton states in quantum mechanics can be conjectured.