Basic tasks of signal processing in spectroscopy - Signal Processing in MRS with Biomedical Applications - page 21

original appearance, Breit and Wigner [90] derived their dispersion formula
or resonance formula (1.3) with the help of a perturbation method of the type
of the WeisskopfWigner theory of dispersion of light by atoms.
A more direct derivation of the BW formula from the viewpoint of a quan
tum mechanical scattering wave function for particle scatterings was made
by Siegert [91]. He also employed the perturbation formalism of quantum
scattering theory. Crucially, his presentation of decaying states of a transient
compound system consisting of a projectile and a target is very plausible. This
is dictated by the nature of the initial problem in which one seeks to describe
an experimental fact that at certain total energies of the compound system,
the cross sections for a collision between a projectile and a target acquire un
usually large values. Such a situation is precisely within the category of the
universal phenomenon called the resonance effect, with similar appearances
irrespective of its origin (mechanical, quantummechanical, acoustic and the
like).
Hence, it is natural to expect that the observed enhancement could be
inferred directly from the notion of the cross section Q(E). By definition,
cross section Q(E), in the units of area for a given collisional event, is given
by a ratio between the outgoing and incoming particle flux Φ (+) and Φ (−)

Q(E)∝ Φ (+)

Φ (−) .

(1.5)

This implies that Q(E)−→∞if Φ (−) −→0. As such, cross section Q(E)
becomes singular with the disappearance of the incoming wave Φ (−) . Hence,
a compound state of the projectiletarget system can be perceived as a state
describing annihilation of the incoming wave. Stated differently, a compound
state is a state whose total energy makes the pertinent cross section singular,
i.e., Q(E) becomes enormously large. Hence the resonance effect. Within this
framework, the target captures (absorbs) the projectile to form a temporarily
existing compound system. These compound states are unstable or metastable
and, thus, prone to decay once the decay time 1/Γ k has elapsed. The inverse
resonance width 1/Γ k is the half lifetime of the k th metastable state. An
alternative way of conceiving this picture could be to say that the compound
system relaxes with the relaxation time 1/Γ k .
This key identification in the definition of Q(E) yields the signature of
the resonance effect as the singularities of the cross sections at certain values
of complex energies E ′ k (k = 1, 2, 3,...) of the compound system. Siegert's

hypothesis of a complex energy E ′ k −iΓ k /2 of the compound system at

the resonance stems from the occurrence that these states possess a definite
width Γ k

= E k

= 0, since they are decaying (radiative) states.
In contrast to genuine or true discrete states of negative energies with zero
widths, resonance states have positive energies. Therefore, a resonance state
is described by a wave packet localized on the positive part of the real en
ergy axis. The real and the imaginary parts of the complex resonance energy