Show that the time-dependent Schrdinger equation preserves the normalization of the wavefunction, i.e. If a function w(x,t) is normalized at t=0, i.e. +00 Sy* (x,0)y (x,0)dx = 1 -00 and y(x,t) satisfies the time-dependent Schrodinger equation, i.e. ih dy(x,t) at dy(x,t) +V(x,t)(x,t), 2m dx then y(x,t) is normalized at any later moment in time t, i.e. +00 Sy" (x,t)y(x,t)dx=1 for any t. -00 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.