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?
Answer
If we take a system, and let it evolve for some indefinite amount of time, it will be in an incoherent mixture of energy eigenstates. Many systems we encounter in nature have been sitting for some time, and not interacting with the environment (much). These can be considered to be in energy eigenstates.
For example, suppose we consider an atom in a gas. Let's assume that the last interaction it had with the environment was a collision with another atom, which put it in some state which we can consider as a superposition of energy eigenstates. Now, let's look at it. Even if we know everything about the collision except how long ago it collided, we can't determine the phase of this superposition, so we might as well think about it as being in a probabilistic mixture of energy eigenstates.
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