Quantum constants of the motion for two-dimensional systems

Abstract

Consider a non-relativistic Hamiltonian operator H in 2 dimensions consisting of a kinetic energy term plus a potential. We show that if the associated Schrödinger eigenvalue equation admits an orthogonal separation of variables then it is possible to generate algorithmically a canonical basis Q, P where P1 = H, P2, are the other 2nd-order constants of the motion associated with the separable coordinates, and [Qi, Qj] = [Pi, Pj] = 0, [Qi, Pj] = Deltaij. The 3 operators Q2, P1, P2 form a basis for the invariants. In general these are infinite-order differential operators. We shed some light on the general question of exactly when the Hamiltonian admits a constant of the motion that is polynomial in the momenta. We go further and consider all cases where the Hamilton-Jacobi equation admits a second-order constant of the motion, not necessarily associated with orthogonal separable coordinates, or even separable coordinates at all. In each of these cases we construct an additional constant of the motion.