Momentum operator

In position representation, momentum is minus i hbar times spatial differentiation.

The spatial translation operator is the unitary generated by the momentum operator. With the convention $U(\mathbf{y})=e^{-i\mathbf{P}\cdot \mathbf{y}}$, a momentum eigenstate of eigenvalue $\mathbf{p}$ acquires the phase $e^{-i\mathbf{p}\cdot \mathbf{y}}$, while $U(\mathbf{y})\varphi (\mathbf{x})U(\mathbf{y}{)}^{-1}=\varphi (\mathbf{x}+\mathbf{y})$.

In one dimension the momentum operator acts on a position-space wavefunction as $p=-i\hslash {\partial }_x$, and $p^2=-{\hslash }^2{\partial }_x^2$.

In position representation, $x$ acts by multiplication and $p_x=-i\hslash {\partial }_x$, so