Free particle nonenergy eigenfunctions of a transformed momentum operator using classical solution variables

The classical solution is used to provide appropriate variables for the quantum-mechanical free particle time-dependent Schrödinger wave equation, which then becomes separable giving nonenergy eigenfunctions of closed form. Solutions derived in this fashion may fail to satisfy the homogeneous Schrödinger wave equation at one point in space/time; if this should occur, an inhomogeneous term is produced and a corresponding restricted solution reduces to the free particle propagator. In general, it is shown that the nonenergy eigenfunctions may also be interpreted as eigenfunctions of a time-independent transformed momentum operator. The similarity transformation, by means of which is constructed the transformed momentum operator, the transformed position operator, and the transformed Hamiltonian, are explicitly determined. The relationship of the nonenergy eigenfunctions to the energy eigenfunctions is also exhibited.