Energy Propagation in a Finite Lattice

The standing-wave normal modes and their frequencies are derived for a finite, one-dimensional harmonic lattice with force-free boundary conditions. The energy-current operator is derived in terms of excitation and deexcitation operators for the standing-wave energy states. The expectation value of the energy current in any such state is identically 0. External driving and load forces on the ends of the lattice are treated as time-dependent perturbations. The perturbed states of the lattice are found. The expectation values of the energy-current operator in such perturbed states coincide with the classical response of the lattice; the energy current is the linear energy density multiplied by the group velocity. This approach is more realistic than the conventional method using cyclic boundary conditions, describing the energy current in terms of running normal modes (momentum states) that can exist only in topologically unrealistic lattices without ends. It is hoped to generalize this work to finite lattices of 2 and 3 dimensions.