# Correspondence principle

Is there a precise mathematical derivation for the correspondence principle for which I can replace $E \rightarrow i \hbar \frac{\partial}{\partial t}$ and $p \rightarrow -i \hbar \nabla$?

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Assumption that in quantum mechanics particles are waves $\exp(-i E t / \hbar + i \vec{p}\cdot\vec{x}/\hbar )$. –  Piotr Migdal Nov 22 '10 at 16:59
$(E,\mathbf p) \mapsto i\hbar(\partial_t, -\nabla)$ is not the commonly known correspondence principle –  KennyTM Nov 22 '10 at 17:30
The basic way to convert between classical and quantum mechanics is to replace a Poisson bracket (in Hamiltonian mechanics) by $i\hbar$ times the commutator. –  Noldorin Nov 22 '10 at 18:41
As far as I know, they are definitions, so they are not really derived mathematically... –  Sklivvz Nov 22 '10 at 19:42