normalising a wave function from Schrodinger's equation

Question

Say $\psi(\vec r,t)$ is a separable solution that satisfies the Schrodinger's equation such that $\psi(\vec r, t)=\phi(\vec r)f(t)$. Then my book said "$\psi$ is normalised if $\int_{\mathbb{R^3}}|\psi|^2d^3x=1$", which is slightly conflicting since usually when we normalising a wave we only consider $\int_{\mathbb{R^3}}|\phi|^2d^3x$ instead and ignore the function of $t$. So where did I make the mistake in terms of my understanding?

Answer 1

When working with separated solutions of Schrodinger's equation, $f(t)$ always takes the form $$ f(t) = e^{-iEt/\hbar}. $$ So $|f(t)|^2 = 1$ for all $t$, and thus $|\psi|^2 = |\phi|^2$.

Answer 2

In quantum mechanics, time evolution is unitary. The time dependent Schrödinger equation is

$$i\hbar \partial_t |\psi\rangle = H|\psi\rangle $$ Where $H$ is the Hamiltonian. Thus

$$ \partial_t \langle\psi|\psi\rangle=i\hbar(\langle\psi|H^\dagger|\psi\rangle-\langle\psi|H|\psi\rangle)=0$$

since $H$ is hermitian. This proves that time evolution doesn't change the norm of the wave function. If your initial wave function $\psi(\vec{r},0)$ is normalized, it will be normalized at every time $t$