Narrow Results

We propose a new method to derive energy-level gap for Hamiltonians in the context of noncommutative quantum mechanics (NCQM). This method relies on finding invariant eigen-operators whose commutators with Hamiltonian are still the operators themselves but with some eigenvalue-like coefficients, which correspond to the energy-level gaps of the systems. Based on this method, only after some simple algebra, we derive the energy-level gaps for several important systems in NCQM, and most of these results have not been reported in literature so far.

By virtue of the invariant eigen-operator method, we obtain the energy-level gap for an arbitrary number of identical one-dimensional, harmonically coupled oscillators.

For some multiatom molecules, which are composed of three types of atoms alternating regularly located, we introduce the quantum mechanical Hamiltonian operator and then search for its vibrating modes by employing the "invariant eigenoperator" method. This method is simpler than the usual diagonalization method in many cases, and may have wider applications in solving some quantum mechanical lattice Hamiltonian models.