quantum mechanics - Can we correctly define momentum operator only by means of position operator and their commutation relation?

In "J.M. Ziman. Electrons and Phonons: The Theory of Transport Phenomena in Solids" the author formally introduces the position (displacement) operator and then defines the momentum operator with the correct commutation relation $$ [\hat{u}_{l}, \hat{p}_{l'}] = i \hbar \delta_{l,l'}$$ between these two. Is such approach formally correct?

Edit: This suggestion is wrong: Can this lead to some degenerate form of "momentum operator"? First that comes to my mind is $$ \begin{aligned}\hat{p}_{l} = -i\hbar\frac{\partial}{\partial u_{l}} \quad \text{if} \quad u < u_{0} \\ \hat{p}_{l} = -i\hbar\frac{1}{u_{l}} \quad \text{if} \quad u \geq u_{0}.\end{aligned}$$ Putting this theoretical construction aside, the core of the question is that if generally such definition of an operator can be done?