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.

中文翻译：

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具有高斯概率密度的一维Pauli问题的精确解和Gerchberg-Saxton解

检索在位置和动量表示中具有给定高斯概率密度的一维量子态的精确相位。找到的保利伙伴的数量取决于位置和动量的海森堡不确定性关系是否饱和。在不饱和的情况下，找到了两个保利伙伴。它们的位置不确定性的时间导数的符号不同。正确的Gerchberg-Saxton算法可以解决相同的问题。它的收敛也取决于不确定性饱和度。