We consider the time-inhomogeneous Markovian jump process introduced by John S. Bell [Phys.Rep. 137, 49] for a lattice quantum field theory, which runs on the associated configuration space. Its jump rates, tailored to give the process the quantum distribution
∣Ψt∣2 at all times
t, typically exhibit singularities. We establish the existence of a unique such process for all times, under suitable assumptions on the Hamiltonian or the initial state vector
Ψ0. The proof of non-explosion takes advantage of the special role of the
∣Ψt∣2 distribution.
Generally, one thinks of a “quantum jump” as the process in which an electron “jumps” between a pair of quantum states, even as the process is treated within perturbation theory. This jump of an electron has remained a key point of conservative (i.e., traditional) quantum mechanics. But, the question of the time dependence of such a transition, e.g. the time for an atom to be ionized by radiation, is somewhat different than this view. A detailed approach in which an incoming wave first polarizes the quantum states and then completes the transition has allowed for a detailed discussion of the smooth transition of the electron from one state to the next. Here, we will discuss the history of the process, and illustrate the approach with the question of “how long does it take for an electron to emit a phonon?” The entire process arises from the proper application of wave mechanics and obviates the need to even consider a discussion of quantum jumps.