What is the mathematical reasoning behind Schrodinger's equation preserving its normalization, with the evolution of time? I am currently in high-school, currently working on a physics research on the normalization of the Schrodinger's equation. I was quite interested on how we can mathematically explain preservation of the SE's normalization with the evolution of time. I was wondering if anyone can help me understand the physical and mathematical side towards answering the question of how SEs even preserves the normalization in the first place. 
I hope someone out there can help me out in me research... Because my teacher isn't giving me anything useful. 
 A: For a time independent Hamiltonian $H$, The wavefunction evolves like
$$|\psi(t)\rangle=e^{-iHt/\hbar}|\psi(0)\rangle$$ 
Using this we see that
$$\langle\psi(t)|\psi(t)\rangle=e^{iHt/\hbar}e^{-iHt/\hbar}\langle\psi(0)|\psi(0)\rangle=\langle\psi(0)|\psi(0)\rangle$$
Therefore if $\langle\psi(0)|\psi(0)\rangle=1$ then for all time $\langle\psi(t)|\psi(t)\rangle=1$
This is true for unitary transformations.
A: To be honest, it is designed to do that for a very profound property that we expect the Nature to follow, time-translation symmetry. Given this symmetry, Wigner's theorem tells us that the time evolution of a state should be described by unitary linear operators and the Schrodinger equation is just that statement. There may exist multiple different such qualified unitary operators and they describe different allowed systems in Nature such as a simple harmonic oscillator or a free-particle, etc. (or, you can say they describe different possible laws for the Nature as a whole if you consider the time evolution operator of the universe itself).
