  The unoccupied states of complex materials are difficult to measure, yet play
a key role in determining their properties. We propose a technique that can
measure the unoccupied states, called time-resolved Compton scattering, which
measures the time-dependent momentum distribution (TDMD). Using a
non-equilibrium Keldysh formalism, we study the TDMD for electrons coupled to a
lattice in a pump-probe setup. We find a direct relation between temporal
oscillations in the TDMD and the dispersion of the underlying unoccupied
states, suggesting that both can be measured by time-resolved Compton
scattering. We demonstrate the experimental feasibility by applying the method
to a model of MgB$_2$ with realistic material parameters.
