Proving the preservation for the norm of a wavefunction

Question

How can we prove the preservation for the norm of the wave-function for a specific hamiltonian (say a spin 1/2 particle) for all times?

Answer 1

You can simply use that the potential is not explicitly time dependent to factor out the time dependence in the time dependent Schrödinger eq. Then you can plug the relation of $\psi(t,x)$ to $\psi(x)$ in to the norm and see what happens.

Answer 2

HINT: We want to show that the normalization of $\psi$ is constant, i.e., $$ \frac{d}{dt} \left[ \int_0^a \psi^* \psi dx \right] = 0 $$ If the box spans $[0, a]$, then $$ \frac{d}{dt} \left[ \int_0^a \psi^* \psi dx \right] = \int_0^a \left[ \psi^* \frac{\partial \psi}{\partial t} + \frac{\partial \psi^*}{\partial t} \psi \right] dx. $$ Now apply the time-independent Schrödinger equation $i \hbar (\partial \psi /\partial t) = H \psi$. The fact that $\psi(0) = \psi(a) = 0$ will be essential to showing that this quantity vanishes.