We now want to know the probability of measuring the eigenvalue a n if a measurement of the observable A is made. The initial information about the system is given in terms of probabilities. The predictions of Quantum Mechanics are probabilistic. A statistical mixture of states is not equivalent to a linear superposition of states. If a system is in a state , then its initial state is exactly known.
Interference effects are absent for a statistical mixture.
The density operator To introduce the density operator, let us first consider a pure state , i.e. not a statistical mixture of states. The initial conditions are completely known. Most often this operator is chosen to be H . Let the initial state of the system at time t be . The system is in a well defined state, but not in an eigenstate of H , if more than one of the are non zero.