Scattering Resonance’s
Breit-Wigner, Feshbach, and Fano Resonance’s
Physics 612 Quantum Mechanics II
Gcina Mavimbela
Physics and Astronomy
Ohio University
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Contents
Page Number
Introduction
Phase Shifts and Breit-Wigner Resonance Formula
Feshbach Resonance
Fano Resonance
The Feshbabch-Fano Partitioning Method
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3
4
5
6
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Introduction
Scattering is one of the indispensable tools in the study of quantum objects. The main
problem of scattering is determining scattering amplitudes. This caries important
information about the scattering center. A common phenomenon that occurs in scattering
is resonance. We set out to investigate what resonances are and their fundamental
difference to bound states. In order to talk about resonances we need to study phase shifts
from which we realize how resonances or bound states come about.
In scattering we have the scattering center which provides a potential, V. A beam of
particles is projected towards the center, the incident flux and since they are free
particles, we have the free Hamiltonian, H 0 . We consider one particle incident on the
scattering center with possible Eigen values of the free Hamiltonian being the momentum
Eigen states (not proper states because of delta normalization).
H0 
p2
2m
H0  p
(1)
p2

p  Ep p
2m
In the presence of the scattering potential these can no longer the solution to the full
Hamiltonian of the system. Analysis shows that the scattering states are a superposition
of a plane wave and an outgoing spherical wave.
x 
 (p ) ( x) 
1
(2) 3 / 2
exp( ipr / ) 

exp( ip.x / )  f E ( ,  )

r
(2)
The pre-factor in the spherical wave term is the scattering amplitude, it depends on the
energy of the incident particle and the potential, V. This condition (boundary) is also
known as the Sommerfeld Radiation Condition.
The result (2) is ok for practical purposes because in general an experimenter would have
the detector outside the range of the potential of the scattering center. The measurable
observable in experiment is the cross-section which is,
d
E'
2
 f E ( ,  )
d
E
(3)
where E ' is the energy of the out going particle.
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If the scattering is elastic, the differential cross-section is just the square of the scattering
amplitude and if the potential is spherically symmetric the scattering amplitude is a
function of energy and theta only.
Phase Shifts and the Breit-Wigner Resonance Formula
We know that energy tells us whether we get a bound state or not, if we negative energy
we get bound states. Positive energy states confined by a positive barrier are known to
tunnel through eventually. So the states are bound for a certain periods and called
resonances or quasi-bound states.
Solving the Schrodinger equation in the regime of the asymptotic condition (and partial
wave decomposition), one obtains an expression for the scattering amplitude that is phase
shifted by some delta.

f E ( ,  )   (2l  1) f l ( p ) Pl (cos  )
(4)
l 0
Where,
f l ( p) 
1 i l ( p )
e
sin  l ( p)
p
(5)
is the partial wave amplitude.
In the low energy limit, the phase shift,  l ( p) is given by
p 0
tan  l 
(l  1)  R 0 ( R)
( pR) 2l 1
l  R 0 ( R) (1  3  5    (2l  1)) 2 (2l  1)
(6)
The gamma term is the logarithmic derivative of the radial solution,
 ( R) 
1
d
Rl , p (r )
Rl , p (r ) dr
(7)
From equation (6), we note that the is a singularity if R 0 (r )  l . In this case we obtain
phase shifts of the form,  l   / 2  n , n integer. In the vicinity of these singularity we
can expand the logarithmic derivative to obtain
R ( pR)  l  ( E  E R )
d (R)
dE
(9)
E  ER
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Where the energies E R are the energies corresponding to the momenta giving us
resonances.
Inserting (9) into (6), we get
tan  l 
1
E  ER
( pR ) 2l 1
(2l  q)!!2 d (R)
dE
(10)
Defining the width at half maximum to be
l 
2( pR ) 2l 1
(2l  q)!!2 d (R)
dE
(11)
The above relations lead to the Breit-Wigner resonance form of the amplitude
pf l  exp( i l )  sin  l 
l / 2
E R  E  il / 2
(12)
So the resonances show up as poles in the amplitude, f l (E ) i.e. where
E  E R  il / 2
(13)
The resonances manifest as peaks in the scattering cross-section. The width at half
maximum is related to the lifetime of the resonance. If it is small then, the peak in the
cross-section is sharper and the resonance is long lived. In this case the pole is closer to
the real axis.
An important question to an experimentalist is whether a peak is a resonance or not. To
answer this question one studies the following diagrams diagram associated to the crosssection. If a peak shows all this characteristics, then it is realized as a resonance state. In
particular, the looping in the argand diagram is important (fig 4.5). If the looping is
counter-clockwise then we have a resonance, otherwise clockwise looping means the
peak is not a resonance.
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Feshbach Resonance
In multi-channel scattering, the problem of forming quasi-bound states (resonances) is
interesting. in that the incident particle can either excite the target or form a quasi-bound
state for a moment. In this case we have a scattering center with several discreet energy
states and an incident particle with a continuum of energy states.
Feshbach resonances can occur when there is coupling between two types of motion, e.g.
an electron scattering off Helium ion (single charge). The incoming electron can excite
the ion to the n=2 state if it has enough energy. If its energy is insufficient it can be
captured to form doubly excited Helium and since the two can exchange energy leading
to the ejection of one electron (this is a resonance).
The simplest case is that of an incident particle interacting with a system with two states,
1 and  2 with corresponding energy  1 and  2 , respectively. If the incident particle
with energy E i interacts with the target in its lower energy state (state 1), the total energy
is E  Ei   1 . If E   2 , then the electron cannot excite the system to the state two but
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can be capture into a Feshbach resonance. The full wave function is described in terms of
the closed and open channels. The closed channel function dies off at large distances
from the scattering center while the open channel function is an oscillatory phase shifted
function at large distances from the scattering center (continuum).
The typical Feshbach resonance has simple properties. The phase shift of the continuum
wave function increases by pi in an energy range covering the resonance, where the
resonance’s energy is dependent on the coupling between the two channels. The
probability of being in resonance, integral of the absolute square of the closed channel
wave function over all r, peaks at resonance energies. The maximum of the probability as
a function of the energy is inversely proportional to energy width at resonance. The
following diagram shows a the potential curves of the open and closed channels as a
function of the separation distance, R between the center and incident particle.
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Fano Resonances
The Fano resonance, like the Feshbach resonance arises in multi-channel scattering as a
result of coupling in the channels. These resonances are observed in atomic and
molecular systems when we have system energy is above the threshold ionization energy.
For instance we can have a helium atom in the state (2s2p) 1 P1 state, whose energy is
above the lowest ionization energy threshold. This state can decay to He   e if excited
and these would manifest as a highly asymmetrical peak in the excitation spectrum. The
following diagram is an illustration of a typical feature of the resonance, the Fano shape.
The Feshbach-Fano Portioning Method
One tool used to analyze multi-channel scattering is a scheme known as the FeshbachFano Partitioning method. In this method with start out with the stationary Schrodinger
equation with the asymptotic condition that the scattering state is composed of a plane
wave and an outgoing spherical wave,
H p  E p
(14)
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

r  p r

1
(2 ) 3 / 2
 i. p .r
e ipr 
e

f


r 

(15)
We choose a pair of operators (complementary), P and Q with the following properties.
Q  P  1; QP  0; Q 2  Q; P 2  P
(16)
and


lim r P  p  lim r  p
r 
(17)
r 
Feshbach’s partitioning of the Schrodinger equation means expanding its solution in this
way,
p  P p  Q p
(18)
with asymptotic conditions

lim r P  p 
r 
1
(2 ) 3 / 2

lim r Q  p  0
 ip .r
e ipr 
e

f

f


P
Q
r 



(19)
r 
The goal is define the partitioned amplitudes in terms of the solution to the background
Schrodinger equation, eq.(20)
PHP  p  E  p
(20)
This leads to the following results for the pieces of the partitioned function,
Q p 
1
QHP  p
E  H eff ( E )
P p   p 
(21)
1
1
PHQ
QHP  p
E  H eff ( E )
E  PHP
'
(22)
Taking the limit of the position representation of both (21)and (22) and comparing with
(19) yields
f Q  4 2   prˆ PHQ
1
QHP p
E  H eff ( E )
(23)
From which we obtain the resonant T matrix
Tres  PHQ
1
1
QHP  PHQ
QHP
E  H eff ( E )
E  QHP  ( E )  i( E ) / 2
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(24)
The solutions (in the lower complex-E plane) to the implicit equation (25) give the
resonances,


det z res  H eff ( z res )  0
(25)
This gives a set of equations to be solved for the poles, z res .
References
Landau R.H (1989) ‘Quantum Mechanics II’ (Wiley & Sons publishers).
Bandopadhyay S, Dutta-Roy B, Mani H.S. (2004) ‘Understanding the Fano Resonance
Through Toy Models’ (Am. J. Phys. 72 (12))
Elster C. (2007) Chapter 11: Elementary Scattering Theory (Ohio University)
Fano U. (1961) Effects of configuration Interaction on Phase Shifts (Phys. Rev 146 (6))
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