It came to me just now that because we can always take the Fourier transform of a well-behaved function, this means we can think of any such state as a superposition of free-particle momentum eigenstates. E.g., the Hermite polynomial eigenfunctions of the harmonic oscillator. They have a Fourier transform (whatever it is) and can therefore be thought of as superpositions of functions e^(ipx) . These are free momentum eigenstates, even though individually they are not solutions to the harmonic oscillator Schrodinger equation. From a math point of view, this is trivial. But does it have any theoretical significance?