# How can a solution of the time-independent Schrödinger equation evolve in space?

I understand that if the Hamiltonian does not depend on the time, the Schrödinger Equation becomes separable, so you get

$$H \psi(x) = E \psi(x)$$

and

$$\Psi(x,t) = \psi(x)\exp\left(-\frac{\imath E}{\hbar}t\right).$$

But $-\frac{\imath E}{\hbar}t$ is a purely imaginary number, so $$\left|\exp\left(-\frac{\imath E}{\hbar}t\right)\right| = 1$$

If that is correct, then how can there be any probability density flow in time? The $exp$ term is only changing the phase of $\psi$, but does not contribute anything to its absolute value.

What did I understand wrong?

What you've written is only true if $\psi(x)$ is an eigenstate of $H$.
For some general $\psi(x)$ that is not an eigenstate (i.e. $H \psi(x) \ne E \psi(x)$), then $\Psi(x, t)$ will be more complicated than just the time independent wavefunction multiplied by a phase factor.