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I am aware that because a particle whose wavefunction we are dealing with must be found somewhere, we normalize the probability density in position. Why do we not normalize the wavefunction in time? given that the wavefunction is a function of both time and space. I am aware that in the case of separable solutions, exponential time dependence is obtained for the wavefunction. But I see that it cannot be normalized for time running from minus to plus infinity as with any exponentially dependent function.

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  • $\begingroup$ Time normalization is already "built in," because the time evolution operator is unitary, and unitary operators conserve total probability. (The exceptions are in cases like the decay of unstable particles, but note also that the normalization of the wavefunction is time-dependent anyway in that case, since the probability of the particle existing somewhere is not necessarily $1$, and decreases with time.) $\endgroup$ – probably_someone May 7 '19 at 12:11