Quantum dynamics of a solvable nonlinear chiral model

Abstract

The quantum mechanical analogue of a classical nonlinear system is shown to be exactly solvable and its energy levels and eigenfunctions are obtained completely. The symmetric version (k₀=0) of this model is the SU(2)(X)SU(2) chiral invariant Lagrangian in the Gasiorowicz-Geffen coordinates. The radial part of the classical equation of motion (in both the symmetric and non-symmetric cases) admits simple harmonic bounded solutions and the bound state energies of the quantized system show a linear dependence on the coupling parameter lambda . It is shown that the Bohr-Sommerfeld quantization procedure reproduces the form of the correct bound state energy levels while a perturbation theoretic treatment gives the exact energy expressions. The ordering problem that arises in the quantum mechanical case is overcome.