Gauge-invariant periodic solutions of the exact time-dependent Schrödinger equation and their time averages

Abstract

The gauge-invariant periodic quantization (GIPQ) method is applied within the complete Hilbert space of solutions of the exact time-dependent Schrödinger equation. The multiplicity of such GIPQ solutions is shown to exceed by far the number of eigenfunctions they are intended to analogize. However, time averaging (or the equivalent averaging over all values of the progressive phase) reduces in this space to a projection upon the exact eigenfunctions under which every one of these GIPQ solutions either vanishes identically or reduces precisely to one of the stationary (but unnormalized) eigenfunctions of the exact Hamiltonian. One concludes that the correspondence between the GIPQ solutions and exact eigenfunctions is well manageable in the complete Hilbert space, a result which encourages the further study of time-averaged GIPQ solutions in parametric subspaces, where time averaging projects not upon an eigenstate, but upon the constant component of the GIPQ periodic solution.