Basically any measurement is on wave function $|\psi\rangle$ is done by operator $X$ such that $X|\psi\rangle$ results observable $x$ with some probability.

But what happens to $|\psi\rangle$? Does it continue to evolve by the same hamiltonian $H$ as it was before? Or does the wave function suddenly change from the previous hamiltonian evolution and evolves differently from what it would have if not measured?

Here I am thinking $|\psi \rangle$ as an objective wave function, encoding probability distribution that would be right in infinite time, not our subjective knowledge of particles.