On the use of Lie-Bäcklund operators in quantum mechanics

AbstractThis paper uses Lie-Bäcklund operators to study the connection between classical and quantum-mechanical invariants and their relations to symmetry and to separation of variables. The problem of an isomorphic correspondence between classical and quantum mechanics is studied concretely and constructively. For functions at most quadratic in the momenta an isomorphism is possible which agrees with Weyl's transform and which takes invariants into invariants. It is not possible to extend the isomorphism indefinitely. The requirement that an invariant goes into an invariant may necessitate variants of Weyl's transform. This is illustrated for the case of cubic invariants. Finally, the case of a specific value of energy is considered