Motivation behind the conjugate Schrödinger equation

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What is our motivation behind the conjugate Schrödinger equation (SE), mathematically a conjugate complex scalar is a "projection" of that complex scalar on its opposite quadrant. But what's going on with SE? I mean, the time-dependent part turns negative, phisically, what does it means? Are we projecting the SE with a "time inversion SE" when we want to find the probability density?