Question

a free particle is moving in x direction with linear momentum P the wavefunction of particle normalized in length L is​

Answer 1

Answer:

No, when you normalize to a length L that only means you should use a region of length L to do the normalization integral

Here x0 can be anything. It shouldn't matter what it is because, presumably, you are assuming the wavefunction is periodic with spatial period L. There is no need to impose any boundary condition other than that.