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 $H=\frac{p^2}{2}+V\left(q\right)\mathrm{}$ 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 $H(p,q)\mathrm{}$ 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.