homework and exercises - Total angular momentum in QFT

Can anyone show explicitly how the QFT total angular momentum operator

$$\hat{\vec{J}} = - i \int \frac{d^3p}{(2 \pi)^3} \hat{a}^{\dagger}_{\vec{p}} ( \vec{p} \times \nabla_{\vec{p}}) \hat{a}_{\vec{p}}$$ gives $$\hat{\vec{J}} |\vec{0} \, \rangle = 0~?$$

Derivation of all the above is here. The question is essentially the same but I can't really implement the existing answer mathematically.

$|\vec{0} \, \rangle$ being a momentum eigenstate.

Answer

If the state you're using is just the vacuum, then my comment to your question applies. If otherwise $|\mathbf 0\rangle = a_{\mathbf 0}^\dagger|0\rangle$, then just use that $[a_{\mathbf p},a_{\mathbf q}^\dagger] = \delta_{\mathbf p,\mathbf q}1$, with $\mathbf q=\mathbf 0$ to fix the integral at the term with $\mathbf p = \mathbf 0$ through the Dirac delta $\delta_{\mathbf p,\mathbf 0}$.