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In first year QM we postulated $\hat p = -i \hbar \nabla$ and used it to derive that $[\hat x, \hat p]= i \hbar$.

In fourth year QM they postulated $[\hat x, \hat p]= i \hbar$ and used spatial symmetry argments to derive the form of the momentum operator $\hat p$.

Am I correct in thinking that in first year we used an 'overkill' postulate of $\hat p$, and that the minimal axioms of QM would use the fourth year order? Or is it really just different conventions?

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Since the commutation relation can be used to derive $\hat{p}$, and the explicit action of $\hat{p}$ in the x basis can be used to verify the commutation relation, you are using an equivalent set of axioms regardless of which one you choose to include. Therefore it is up to the author's pedagogical perspective which one to start with.

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  • $\begingroup$ This could be obvious, but are there therefore no alternative forms of $\hat p$ that could satisfy the commutation relation? $\endgroup$
    – Alex Gower
    Oct 12 at 12:48
  • $\begingroup$ Ah I see from one of the comments that there is probably a unitary transformation connecting these alternative forms. $\endgroup$
    – Alex Gower
    Oct 12 at 12:53
  • $\begingroup$ So ultimately, you can equivalently choose to fix the form of $\hat p$ or the cannonical commutation relation, and neither is necessarily a 'better' axiom choice, agreed? $\endgroup$
    – Alex Gower
    Oct 12 at 12:53
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    $\begingroup$ Agreed! At least not in a logical sense. The only argument you can make is pedagogically but that is physics pedagogy and not physics itself. $\endgroup$ Oct 12 at 13:12
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    $\begingroup$ @AlexGower The fact that position and momentum operator are (up to unitary equivalence) uniquely determined by the canonical commutation relations is called the Stone-von Neumann theorem. $\endgroup$
    – Noiralef
    Oct 12 at 19:21

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