Is there a function that describes how a collapsed wavefunction (particle or delta function?) is restored back into a wave?

Question

The moment an electron is observed (interaction), the electron takes on particle properties, but it is thought that it will recover its wave properties over time.

Even if an electron becomes an existence occupying a local area like a particle for a moment through some observation or interaction, shouldn't it be restored to an electron with wave properties over time?

Is there a function that describes how a collapsed wavefunction (particle or delta function?) is restored back into a (non-collapsed) wave?

Answer 1

Yep, it's called the Schrodinger equation! The Schrodinger equation states that if $H$ is the Hamiltonian, or total energy operator of a particle with wavefunction $\Psi$, then$$i\hbar\frac{\partial\Psi}{\partial t}=H\Psi.$$In other words, the Hamiltonian operator (which multiplies each basis state wavefunction by the corresponding energy of that wavefunction) also describes the time derivative of the wave function (how it changes). So if a particle is measured at $t=0$, then at that point its wavefunction suddenly becomes $1$ at a single point and $0$ at all other points. From there, the Schrodinger equation gives you the time derivative of the wavefunction, which shows how it will evolve and turn back into a delocalized probability density function again.