Quantum field theory is "just" a specific kind of quantum mechanics. Philosophically, you can think of relativistic quantum field theory in different ways that lead you to the same place, such as:
- A quantum theory of an indefinite number of identical particles (which we need to describe interacting relativistic particles),
- A quantum theory where the fundamental degrees of freedom are fields living on spacetime.
- A low energy description of local interactions with a given set of symmetries (aka, Wilsonian effective field theory).
Given that quantum field theory is just "a type of" quantum mechanics, anything you can do in quantum mechanics, you can also in quantum field theory, at least in principle.
The main reason that scattering calculations are emphasized in field theory books, is that scattering experiments are the primary source of experimental data, for physics involving relativistic particles. Also, there is a well established perturbative framework for computing scattering amplitudes, that forms the basis for more sophisticated techniques.
The main reason to use fields, as opposed to a Hamiltonian density, is that fields encode local interactions. In relativity, non-local interactions would lead to a non-causal theory, so it is important that interactions occur at well defined points in spacetime. A Lagrangian built out of fields encodes locality. There is more about this in Weinberg, Volume 1, under the guise of the "cluster decomposition principle."
However, there are other quantities and effects that can be calculated in field theory, beyond scattering, including:
- Power spectra of inflationary perturbations, that (if inflation is true) become the initial seeds of structure in the Universe and which we observe in the cosmic microwave background
- Properties of phase transitions (when field theory is applied to condensed matter systems, which are usually not relativistic)
- Tunneling effects, including the probability for false vacuum decay to occur
- Asymptotic freedom and confinement (the latter needs experimental data + lattice QCD to justify)
- The Lamb Shift of Hydrogen
The main issue with quantum field theory, compared to quantum mechanics of a fixed and relatively small number of particles, is that it is complicated to handle many particles. (Incidentally, that's not to say that quantum mechanics of a small number of particles is easy -- quantum mechanics problems with interactions between a few particles can quickly become intractable). The analogue of a few line calculation in single particle quantum mechanics may take many pages in quantum field theory. Part of this is kind of a trivial complication of dealing with more degrees of freedom However, part of this is "physically deep" in the sense that stuff can happen in quantum field theories that don't happen in quantum theories of a fixed number of particles, such as spontaneous symmetry breaking in the ground state. Books on field theory reflect this by quickly becoming very technical as they need more sophisticated tools to make sense of calculations, which are overkill in non-relativistic quantum mechanics.
Anyway, this is all to say that:
- Quantum field theory is a "type" of quantum mechanics. Anything you can do in a general quantum theory (such as describe the time evolution of a state), you can also do in a quantum field theory.
- Books focus on scattering for historical and experimental reasons, not logical ones. (Meaning, there's nothing about quantum field theory that says you logically must study scattering amplitudes, not that experiments aren't important, since of course they are).
- Fields are used because they encode locality of interactions.