Phase-space Propagation of the Time-independent Schrödinger Equation for One-dimensional Rectangular Potential Systems

A thorough analysis of one-dimensional rectangular potential systems, encompassing both barriers and wells, is provided within the framework of the phase-space propagation method for the Time-Independent Schrödinger Equation (TISE). An initial value representation approach is adopted, involving an ensemble of initial conditions invariant under phase-space flow on the left side of the barrier or well. The system’s periodicity in the free-motion regions, coupled with the simplicity of the potential, facilitates the derivation of analytical expressions for the evolution of the phase-space state across the barrier or well and onto the right side. The transfer matrix method is employed to obtain explicit analytical expressions for the phase-space state of the one-dimensional barrier and its transmission coefficient. A similar approach is extended to the potential energy well, where transcendental equations for both bound and virtual (anti-bound) states are derived through a geometrical analysis of the phase-space flow of the TISE and the stable and unstable manifolds. Furthermore, a detailed analysis of the transmission coefficient is conducted. It is shown that the energy curves for the nth bound state and the \((n+2)\)th virtual state tend toward the same asymptotic limit. This limit corresponds to the energy curve (or line) of the \((n+1)\)th resonance, offering deeper insights into the system’s energy spectrum and its behavior in the limit of infinite potential wells.