Momentum operator

\overset{p}{^} = - i ℏ \frac{\partial}{\partial x}

In three dimensions In position representation, we define it by using the gradient, and so we see that

\overset{p}{^} = - i ℏ \frac{\partial}{\partial x}

Position and Momentum from Wavefunctions by Faculty of Khan (2018)

Source. Proves in detail why the momentum operator is

\frac{\partial}{\partial x}

. The starting point is determining the time derivative of the expectation value of the position operator.

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