In interacting real scalar field theory, if I intuitively define the "wave function" of a state as
$$\Psi(x)\equiv\langle\Omega|\hat{\phi}(x)|\Psi\rangle.$$
Does this wave function satisfy classical wave equation by letting the variation of action $S$ be zero? The wave equation shall reduce to KG equation in free theory.
Also, the IN state in the interaction theory is see this link
$$\hat{a}_{k}(-\infty)|\Omega\rangle=|k\rangle.$$
How would I intuitively understand this equation? Can I say that the wave functions of the two states are almost the same when $t=-\infty$? If this is correct, can I write the following equation?
$$\langle\Omega|\hat{\phi}(-\infty,\vec{x})\hat{a}_{k}(-\infty)|\Omega\rangle=\langle\Omega|\hat{\phi}(-\infty,\vec{x})|k\rangle.$$
