Position Basis Representation of the Quantum Mechanical Momentum Operator

As an undergraduate, I found the form of the momentum operator in quantum mechanics $ \hat{p}_x = -i\hbar \frac{\partial}{\partial x}$ a mystery. This talk presents a derivation, suitable for undergraduate students, of this well known position-basis representation of the momentum operator. The derivation utilizes standard canonical variable commutation relationships and the Dirac delta function with its derivative to explain how the form of this operator naturally arises in quantum mechanics. This work was inspired by a series of lectures on quantum mechanics given by Prof. Balakrishnan at IIT Madras.