As far as I'm aware you can solve for the wave functional $\Psi[\phi]$ of a field using the Schrodinger equation $\frac{\partial \Psi}{\partial t}=-iH\Psi$.

1. Should $H$ here be the Hamiltonian, or the Hamiltionian density?
2. If it's the Hamiltionian, is there a version of this equation for the Hamiltonian density?
3. Would assuming a certian form for the funcional like $\Psi\sim e^{-\omega \phi^2}$ be of any use in simplifying the equation?

I'm pretty new to QFT so excuse me if the question is poorly formulated.

As far as I'm aware you can solve for the wave functional $\Psi[\phi]$ of a field using the Schrodinger equation $$i\hbar\frac{\partial \Psi}{\partial t}=H\Psi.$$

1. Should $H$ here be the Hamiltonian, or the Hamiltonian density?
2. If it's the Hamiltonian, is there a version of this equation for the Hamiltonian density?
3. Would assuming a certain form for the functional like $\Psi\sim e^{-\omega \phi^2}$ be of any use in simplifying the equation?

I'm pretty new to QFT so excuse me if the question is poorly formulated.