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Yes,
$$|\phi \rangle = \hat{\vec{p}} |\psi \rangle$$
is a correct notation. Note that these are three vectors, because ##\hat{\vec{p}}## are the three operators for the three components of momentum.
$$|\phi \rangle = \hat{\vec{p}} |\psi \rangle$$
is a correct notation. Note that these are three vectors, because ##\hat{\vec{p}}## are the three operators for the three components of momentum.