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I'm trying to convince myself that momentum is the correct infinitesimal generator for translations.

In his book, Sakurai justifies this assertion by making an analogy between the translation operator acting on the ket-space [itex]\mathcal{T}(d\mathbf{x}) : \mathcal{H} \to \mathcal{H} ; | \mathbf{x} \rangle \mapsto | \mathbf{x} + d\mathbf{x} \rangle[/itex] and the type-2 generating function which gives the canonical translation of coordinates.

Is there any stronger mathematical justification other than ``they look similar'', or is this one of those fundamental postulates that can only be proven by experiment?

Thanks.

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# Momentum as the generator of translation

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