Lower bound to the ground-state expectation value of a positive unbounded operator using related bounded operators

摘要

Error bars around observables of quantum mechanical systems are extremely lacking; in most cases only an upper bound to the energy is practical. We present a new lower bound to the expectation value of an operator that is most similar to the lower bound of Weinhold. While Weinhold’s bound has flexibility by incorporating expectation values (some of which may not exist) of different moments of the operator to be bounded, the flexibility of our lower bound relies on the form of a similar, but bounded, operator. Like Weinhold’s bound, ours is limited to non-negative operators and the ground-state of the system. Our lower bound is shown to have properties which allow it to converge to the true expectation value of the ground state, but a practical application to the Helium atom shows that Weinhold’s bound is superior in this case.