Quantum Mechanics for Scientists and Engineers

So far in the treatment of creation and annihilation operators, we have considered operators, including especially Hamiltonians, in which we are concerned with only one kind of particle (i.e., either identical bosons or identical fermions). Many important phenomena involve interactions of different kinds of particles (e.g., interactions of photons or phonons with electrons). Here, we discuss how to handle operators for such situations. As a specific, and particularly useful example, we discuss the electron–photon interaction. This leads us through perturbation theory in this operator formalism to a proper quantum mechanical treatment of absorption and stimulated and spontaneous emission.

States and commutation relations for different kinds of particles

The approach is an extension of what we have done before. We need two additions. First, though we continue to work in the occupation number representation, we must include the description of the occupied single-particle states for each different particle in the overall description of the states. Second, we need commutation relations between operators corresponding to different kinds of particles.

In considering the occupation number basis states – for example, for a system with two different kinds of particles – we simply have to list which states are occupied for each different kind of particle. Suppose that we have a set of identical electrons and a set of identical bosons (e.g., photons).