Can the Born rule be derived? [duplicate]

This question already has an answer here:

$\renewcommand{ket}{|#1\rangle}$ If we have a particle and we know the initial state $|\psi\rangle$ of everything that is relevant, and we know the full Hamiltonian $H$, then we should be able to show that if we measure the position of the particle then we will find it in a position eigenstate, or if not a position eigenstate, at least a superposition of eigenstates in which the observer observes the particle to be in a position eigenstate. From this it would seem as though the Born rule might be able to be derived. Does this reasoning work? If it doesn't work, why not?

Edit: To be clear, it doesn't necessarily have to follow from unitary time evolution by itself as touched in this other question. Suppose $\ket{\psi(0)} = \ket{E}\ket{M}$ where $\ket{E}$ is an energy eigenstate describing the particle and $\ket{M}$ is the state of the measuring device or the environment or whatever is relevant. If one were to now measure the energy of the particle, then knowing the Hamiltonian $H$ (or maybe it is time dependent $H(t)$) it seems that it should be implied, by Schrodinger's equation, that $\ket{\psi(t)}=\ket{E}\ket{M(t)}$, and so the particle is still in the same energy eigenstate. If this is the case then it seems like the Born rule may also be a consequence of the other postulates of quantum mechanics. Is there something not right about what I've said?