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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
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$(E,\mathbf p) \mapsto i\hbar(\partial_t, -\nabla)$ is not the commonly known correspondence principle –  KennyTM Nov 22 '10 at 17:30
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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
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As far as I know, they are definitions, so they are not really derived mathematically... –  Sklivvz Nov 22 '10 at 19:42
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up vote 5 down vote accepted

There are a few different ways to establish this correspondence, eg, using Lie groups or Fourier Transform. But, in the end of the day, the notion that this takes one from Classical to Quantum Mechanics is nothing but an 'axiom'. So, in this sense, it's a bit weird to use the word "derive" the principle: this 'principle' is used as one of the axioms that defines Quantum Mechanics.

In any case, von Neumann was probably the one to first formalize this construction in terms of what he called "Transformation Theory", which is the theory of Fourier Transforms for distributions (generalized functions).

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