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=p^{2}2+V (q) into a set of operator difference equations on a lattice. The equal-time commutation relations are exactly preserved and thus are consistent with the requirements of unitarity. We consider general Hamiltonians of the form H (p, q) and 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 result shows that it is possible to treat quantum spin systems by the method of finite elements.