Resolution of the operator-ordering problem by the method of finite elements

The method of finite elements converts the operator Heisenberg equations that arise from a Hamiltonian of the form H = P2/2+V(q) into a set of operator difference equations on a lattice. These difference equations exactly preserve the equal-time commutation relations and thus are consistent with the requirements of unitarity. In this paper we consider general Hamiltonians of the form H(p,q). We show that the requirement of unitarity uniquely determines the operator ordering in such Hamiltonians. (The ordering procedure involves a set of orthogonal polynomials which are not widely known.) Our results shows that it is possible to treat quantum spin systems by the method of finite elements