Is commutation relation $[\hat x, \hat p]= i \hbar$ or momentum operator $\hat p = -i \hbar \nabla$ an axiom of QM?

Question

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?

Answer 1

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.