There are two sources of 'waviness" in quantum field theory: waves in the underlying classical field, and the Schrodinger equation. I'm learning QFT using the notes by David Tong (I believe they are pretty standard), and having hard time reconciling the two notions even in the example of the free scalar field.
It is mentioned that quantum states in this context are functions $\Phi:\{\mathbb{R}^4 \to \mathbb{R}\} \to \mathbb{C}$ analogous to wave functions in classical QM. Or perhaps they should be $\Phi_t:\{\mathbb{R}^3 \to \mathbb{R}\} \to \mathbb{C}$, as to have the Schrodinger picture going. I assume the latter and drop the subscript $t$.
Then quantum fields are defined as fields of operators acting on the states, in particular: $\hat \phi (\vec x)$ and $\hat \pi (\vec x)$ (I added the hats for clarity).
What is $\hat \phi (\vec x)\Phi$ equal to? Is it defined by $(\hat \phi (\vec x)\Phi)(\phi)=\phi(\vec x) \Phi(\phi)$ for a field configuration $\phi$?
Then creation/annihilation ($\hat a_{\vec p},\hat a_{\vec p}^\dagger$) operators are used to find eigenstates of the Hamiltonian operator in terms of $|0\rangle$. In particular, the one-particle eigenstate of the Hamiltonian with momentum $\vec p$ is $|\vec p\rangle = \hat a_{\vec p}^\dagger|0\rangle$.
I saw the following description of $|\vec p\rangle$ as a wave function. Let $\phi_{A, \vec p}$ be the classical plane wave at time $t=0$ in the direction of $\vec p$ with frequency proportional to $|\vec p|$ and amplitude $A$. Then $|\vec p\rangle$ ($\phi_{A, \vec p}$) as a function of $A$ is the same as the $n=1$ solution of the simple harmonic oscillator as a function of $\ x$.
Is $|\vec p\rangle$ equal to zero on any other field configuration? What about the values of $|\vec p\rangle$ on shifted $\phi_{A, \vec p}$? (i.e. for the same wave at later times as a classical wave). Perhaps $|\vec p\rangle$ has values $e^{i s}$ (multiplied by the same function of $A$) for the shifted by $s$ wave and zero for anything else?
What about $|\vec p\rangle_t$? I want to say that the unitary transformation $e^{-i \hat H t \over \hbar}$ "propagates" the classical wave by shifting the probability amplitude to the shifted $\phi_{A, \vec p}$ to classically later times, but not at all sure if that's true. Can something like that be said in general, or at least for Hamiltonian eigenstates? That is, can the Schrodinger equation "propagate" quantum states in this classical sense? More generally, what about wave packets composed of different momenta?
What is $|0\rangle$ as a wave function?
