Exact and Gerchberg-Saxton solutions of the one-dimensional Pauli problem with Gaussian probability densities

The exact phase of one-dimensional quantum states with given Gaussian probability densities in the position and momentum representations is retrieved. The number of Pauli partners that are found depends on whether the Heisenberg uncertainty relation for position and momentum is saturated or not. Without saturation, two Pauli partners are found. They differ in the sign of the time derivative of their position uncertainties. The same problem is solved by an exact implementation of the Gerchberg-Saxton algorithm. Its convergence depends on uncertainty saturation as well.