Show that the time$-$dependent Schr$$dinger equation preserves the normalization of the wavefunction, i$.$e$.$

If a function $(x,t)$ is normalized at $t=0,$ i$.$e$.$

${\displaystyle \underset{-}{\overset{+}{}}}{}^{**}(x,0)(x,0)dx=1$

and $(x,t)$ satisfies the time$-$dependent Schrodinger equation, i$.$e$.$

$i\frac{\text{d}\text{e}\text{l}(x,t)}{\text{d}\text{e}\text{l}\text{t}}=-\frac{{}^2}{2m}\frac{d^2(x,t)}{d{x}^2}+V(x,t)(x,t),$

then $(x,t)$ is normalized at any later moment in time $t,$ i$.$e$.$

${\displaystyle \underset{-}{\overset{+}{}}}{}^{**}(x,t)(x,t)dx=1,$ for any $t.$

Note: it is possible to prove this even for an arbitrary time$-$dependent potential energy $V(x,t).$ Thus the wavefunction that satisfies the time$-$dependent Schrodinger equation automatically obeys the normalization condition.

Hint: Calculate the time derivative of the normalization integral and prove that it is zero.