Classical versus quantum integrability

Abstract

Classical integrability and quantum integrability are compared for two degrees of freedom Hamiltonian systems. We use c-number representatives for quantum operators and the Moyal bracket for the commutator. Three different cases are found: (i) the c-number representative of the quantum mechanical second invariant is identical to the classical second invariant, (ii) O(ℏ2) corrections are needed in the classical second invariant to obtain the quantum invariant, and (iii) also the potential must be deformed by an O(ℏ2) term. Several examples from the Henon–Heiles and Holt families of integrable potentials are included.