Decay of a Discrete State into a Continuum of States: Fermi’s Golden Rule

Abstract

The last topics I will cover are Fermi’s Golden Rule and irreversible decay. These are pretty interesting topics that involve situations in which a discrete state of a quantum system is coupled to an infinite continuum of quantum states. Although one starts with a Hermitian Hamiltonian, there is irreversible behavior in the system. This is approximately true in many cases of practical interest. For example, an atom that is prepared in an excited state decays as a result of its interaction with the continuum of vacuum field modes. A hydrogen atom placed in a high frequency optical field is photo-ionized into a continuum of (Coulomb modified) free-particle states. The reason we can get irreversible behavior in these cases is connected with the fact that the quantization volume goes to infinity. This feature in already seen in the quantum revivals of a wave packet in the infinite square square well discussed in Chap. 6 There are always quantum revivals, but the revival time goes to infinity as the well size approaches infinity. In other words, an outgoing wave packet can never be reflected back by the boundary of the quantization volume. In problems of this nature, I start by writing equations in which all states are discrete and then take the limit in which some of the states form a continuum.