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$\renewcommand{ket}[1]{|#1\rangle}$ 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?


marked as duplicate by John Rennie, ACuriousMind, Kyle Kanos, Martin, user10851 Jul 24 '15 at 15:41

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    $\begingroup$ possible duplicate of Born rule and unitary evolution $\endgroup$ – John Rennie Jul 24 '15 at 6:32
  • $\begingroup$ I'm a bit surprised this has attracted two downvotes. It's a perfectly reasonable question, and although it's a duplicate the duplicate wasn't that easy to find. $\endgroup$ – John Rennie Jul 24 '15 at 9:39
  • $\begingroup$ @JohnRennie Not sure about who downvoted or why, but just searching "Born Rule" finds the duplicate. $\endgroup$ – Omry Jul 24 '15 at 11:32
  • $\begingroup$ @JohnRennie Actually I did see the other link and I thought that either my question had a subtle difference or that it was unresolved. I am not interested in a derivation of the Born rule that follows necessarily from unitary time evolution, only that it follows from something. Prathyush's comment on the original post describes more or less how I am feeling about it. $\endgroup$ – JLA Jul 24 '15 at 16:13
  • $\begingroup$ How does Akhmeteli's answer not address your concerns? $\endgroup$ – Kyle Kanos Jul 24 '15 at 18:05