In Chap. 2 we solved for the probability amplitudes Ca and C.b explicitly. The results of our discussion, such as the probability of a transition or the value of the induced dipole moment, were invariably expressed in terms of bilinear combinations of the amplitudes, such as CaCa* and CaCi>*. In fact, the expectation value of any observable involves bilinear combinations. Hence an alternative formulation of quantum mechanics consists of dealing with the bilinear quantities directly. The method involves organizing the quantities in a matrix form called the density matrix. There are two principal advantages in doing so. First, the resulting mathematics is often simpler. Sec­ ond, when the wave function for an ensemble of systems is not known but the probabilities for having various different wave functions are known, the en­ semble can be described by a weighted sum of individual density matrices. In particular, the amplitudes cannot themselves describe simply certain com­ mon statistical phenomena, such as the effects of elastic collisions on the in­ duced dipole moments of atoms.