In Matthew Schwartz's QFT text, he derives the Schrodinger Equation in the low-energy limit. I got lost on one of the steps.
First he mentions that
$$ \Psi (x) = <x| \Psi>,\tag{2.83}$$
which satisfies
$$i\partial _t\Psi(x)=i\partial_t< 0|\phi (\vec{x},t)|\Psi>=i<0|\partial_t\phi(\vec{x},t)|\Psi>.\tag{2.84}$$
That was all fine and good, but he lost me on the next part, going from the first line (i) to the second (ii).
(i)$$i<0|\partial_t\phi(\vec{x},t)|\Psi>=<0|\int \frac{d^3p}{(2\pi )^3} \frac{\sqrt{\vec{p}^2+m^2}}{\sqrt{2\omega _p}}(a_pe^{-ipx}-a_{p}^{\dagger}e^{ipx})|\Psi>$$ (ii)$$=<0|\sqrt{m^2-\vec{\nabla}^2}\phi_0(\vec{x},t)|\Psi>.\tag{2.85}$$
He apparently uses:
$$\partial _{t}^{2}\phi_0=(\vec{\nabla} ^2-m^2)\phi_0$$
to get the following term
$$\sqrt{m^2-\vec{\nabla}^2}$$
in equation (ii) above, but not quite sure how. Can anyone help me out?
This is on page 24 for those that have the text.
