Determination of occupation probabilities from time-averaged position distributions

We show that the occupation probabilities of the energy eigenstates excited in a wave packet moving in an arbitrary one-dimensional potential can be determined directly from the time-averaged position distribution. The sampling functions are the derivative of the product of the usual eigenfunctions and the linearly independent (non-normalizable) solutions of the Schr\"odinger equation for the same energy eigenvalue. This is the same structure as those for the harmonic-oscillator case.