Real wavefunction from Generalised Hamiltonian Schrodinger Equation in quantum phase space via HOA (Heaviside Operational Ansatz): exact analytical results

This note reports the decomposition of a complex wavefunction into real and imaginary parts via the Heaviside Operational Ansatz (HOA) in the Quantum Phase Space Representation; this for a Complex Schrodinger Equation (CSE) generated by a given Generalised Hamiltonian. Consequently, the CSE is re-cast as a two-component Real Schrodinger Equation (RSE) system that couples the real and imaginary parts of the complex wavefunction from the original CSE. By way of the HOA, the resulting real and imaginary parts of the complex wavefunction may be expressed in terms of each other: the ‘upshot’ being the de-coupling of the two-component system into a separate new RSE for the real part and a new separate RSE for the imaginary part, of the original CSE complex wavefunction. It follows that the new de-coupled RSE for the real (respectively imaginary) part of the original complex wavefunction is ‘informationally-equivalent’ to the new de-coupled RSE for the imaginary (respectively real) part of the original complex wavefunction. Moreover, each of these RSEs for the aforementioned real wavefunctions is also ‘informationally-equivalent’ to the original CSE and it’s complex wavefunction. What’s more, consequent of HOA, the real wavefunctions each have exact analytical expression in quadrature (integral form).