Feshbach resonance

A Feshbach resonance can arise when there is a coupling between two types of motion. For example, suppose an electron scatters off of He+. The incoming electron can excite the He+ ion to an n=2 state and if it does not have enough energy it can be temporarily captured into a resonance state to form doubly excited He. This is a resonance state because, of course, the two electrons can later exchange energy again and one electron will be ejected. The simplest case to consider is when the incident particle interacts with a system that has two states, Phi_1 and Phi_2. Lets denote their energies by e_1 and e_2. If the incident particle has an energy E_1 and the target is in state 1, then the total energy E = E_1+e_1. The incident particle can be captured into a Feshbach resonance if E<e_2 because then it does not have enough energy to excite the system. The full wave function can be written as Psi_E=y_open(r)Phi_1+y_closed(r)Phi_2. We know that the y_open(r) function is a continuum wave which has the form y_open(r)=sqrt(2/pi k)sin(kr+phase) at large r with the wave number k defined by E_1=(hbar k)^2/2M; this function is designed to have an incoming/outgoing particle flux that is independent of energy. The y_closed(r) function goes to zero as r goes to infinity because the incident particle does not have enough energy to excite the system.

Typical Feshbach resonances have certain simple properties. The first is that the phase shift of the continuum wave function increases by pi over an energy range covering the resonance. The resonance has an energy width that depends on the coupling between the channels.  The "probability" for being in the resonance, integral of (y_closed)^2 over all r,  peaks at the resonance energy; the maximum of this probability as a function of energy is inversely proportional to the energy width of the resonance.

This is one frame from the movie. It shows the wave function in the open channel at one energy in the bottom graph, the wave function in the closed channel in the middle graph, and the energy dependence of the phase shift in the open channel in the top graph. The asterisk in the top graph shows which energy the frame is at. The movie shows a sequence of these frames with the only change being that the energy is incremented by a small amount in each successive frame. There are a number of interesting features to notice in the movie. The first is that for energies away from the resonance, -0.0061 or -0.0050, the wave function in the closed channel is small; it is only near the resonance energy that the closed channel function is large. The size of the wave function in the closed channel is proportional to the energy derivative of the phase. Notice the correlation between the wave function in the open channel and in the closed channel. Apparently, the continuum wave gets pulled in by one wave length when the energy increases through the resonance; this is the meaning of the phase increasing by pi.
Feshbach resonance movie, 1.5 Mb