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

Abstract

The method of finite elements converts the operator Heisenberg equations that arise from a Hamiltonian of the form $H=\frac{{p}^{2}}{2}+V(q)$ 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.