Energy eigenstates provide a convenient basis for solving quantum mechanics problems, but they are by no means the only allowable states. Yet it seems to me that particles/systems are assumed to be in energy eigenstates "in nature".
Some examples of what I mean:
- Solving the Schrodinger equation for the Hydrogen atom gives the standard $|n,l,m \rangle$ basis of energy/angular momentum eigenstates. But we speak of "filling up" these orbitals with electrons, or transitions between energy levels. Why should we expect to find the electrons only in such energy eigenstates, as opposed to say, some arbitrary superposition?
- In quantum statistical mechanics we have Bose-Einstein and Fermi Dirac distributions that give the number particles in a state of energy $\epsilon$, but why must a particle be found in a state of definite energy to begin with?