Proof that the probability 1 is conserved by the time evolution:

It can be derived directly from the Schrödinger equation.

Bibliography:

quantummechanics.ucsd.edu/ph130a/130_notes/node127.html from quantum physics by Jim Branson (2003).
That proof also mentions that if the potential V is not real, then there is no conservation of probability! Therefore the potential must be real valued!

Probability current

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