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