Derivation of QM limit of QFT in "QFT and the SM" by Schwartz

[Hill]

TL;DR

How the time derivative moved into the braket?

In this derivation, a basis of one-particle states $\langle x|=\langle \vec x,t|$ is expressed with the field operator, $$\langle x|=\langle 0| \phi (\vec x, t)$$
"Then, a Schrödinger picture wavefunction is $$\psi (x)=\langle x| \psi \rangle$$
which satisfies $$i \partial _t \psi (x) = i \partial _t \langle 0| \phi (\vec x, t)|\psi \rangle = i \langle 0| \partial _t \phi (\vec x, t)| \psi \rangle$$

I need help to understand why the time derivative is applied to the field $\phi$ and not to the state vector $\psi$.

 

[Demystifier]

I need help to understand why the time derivative is applied to the field $\phi$ and not to the state vector $\psi$.

Because the equation is written in the Heisenberg picture, where observables depend on time and state does not depend on time.

 

[Hill]

Because the equation is written in the Heisenberg picture, where observables depend on time and state does not depend on time.

Thank you.