Phase Space Approach to Solving the Schrödinger Equation: Thinking Inside the Box

One of the striking parallels in the development of the method in quantum mechanics and signal processing is that the method never became mainstream in either community. A key reason is undoubtedly the problems encountered in converging the method, problems reported independently in both fields. This chapter presents a theory that optimizes the phase space localization to obtain the most compact possible representation. It shows applications to shaped femtosecond pulses, to the solution of time‐independent and time‐dependent quantum mechanical problems, and to audio and image compression. The chapter then discusses the application of the periodic von neumann basis with biorthogonal exchange (pvb) method to quantum mechanics. Finally, it discusses the application to the time‐independent Schrödinger equation (TISE) and the application to the time‐dependent Schrödinger equation.