The linear equations can easily be generalized to describe a statistical mixture of states and express all physical predictions about such a system in terms of a density operator. All off-diagonal elements are zero. Then is diagonal. In experiments, we are often dealing with systems about which we have incomplete information. The C ions have been stripped of most of their electrons by having been past through a thin foil.
Most ions are in their ground state, but some are in an excited, metastable state.
We are dealing with a statistical mixture of states . For such a system we introduce the density operator and the density matrix with matrix elements . The linearity of implies that . It can be zero even if is not zero. In a statistical mixture of states the averaging can cancel out interference effects.