Quantum Mechanics and a Completely Integrable Dynamical System

Abstract

From the eigenvalue equation (H0 + λV) \ ψn(λ)) = En (λ) | ψn( λ ) ) one can derive an autonomous system of first order differential equations for the eigenvalues En (λ) and the matrix elements Vmn(X) = ( ψm(λ) \ V \ ψn(λ), where λ is the independent variable. If the initial values En (λ = 0) and ψn (λ = 0) are known the differential equations can be solved. Thus one finds the “motion” of the energy levels En(λ). Here we give two applications of this technique. Furthermore we describe the connection with the stationary state perturbation theory. We also derive the equations of motion for the extended case H = H0 + λ\ Vt + λ2 V2. Finally we investigate the case where the Hamiltonian is given by a finite dimensional symmetric matrix and derive the energy dependent constants of motion. Several open questions are also discussed.